MEGARITHS AJANASIOSJEWRIA DIASTASEWNKAIKAJOLIKOI QWROIDIDAKTORIKH DIATRIBH

PANEPISTHMIO PATRWNTMHMA MAJHMATIKWNPATRA 2010

Prìlogo H didaktorik  mou diatrib  ekpon jhke sto Tm ma Majhmatik¸n tou Panepisthm�ouPatr¸n me Trimel  Sumbouleutik  Epitrop  apoteloÔmenh apì ton k. Dhm trio Gewr-g�ou, Anaplhrwt  Kajhght  tou Tm mato Majhmatik¸n tou Panepisthm�ou Patr¸n w epiblèponta kai mèlh aut  , tou kk. StaÔro Hli�dh, Omìtimo Kajhght  tou Tm mato Majhmatik¸n tou Panepisthm�ou Patr¸n, kai Bas�leio Tz�nne , Kajhght  tou Tm mato Majhmatik¸n tou Panepisthm�ou Patr¸n.Apofasistikì rìlo gia thn ekpìnhsh kai suggraf  th didaktorik  mou diatrib  èpaixe h makroqrìnia kai suneq  sunergas�a pou e�qa me tou k. Dhm trio Gewrg�ou kaik. StaÔro Hli�dh. H sunergas�a aut  xek�nhse apì to deÔtero èto twn proptuqiak¸nmou spoud¸n me seir� seminar�wn se eidik� jèmata Genik  Topolog�a .Euqarist¸ jerm� ton k. Dhm trio Gewrg�ou gia thn parìtruns  tou, thn kajod ghs tou kai gia ti polÔtime sumboulè kai upode�xei tou ìla aut� ta qrìnia th episthmo-nik  sunergas�a ma . Ep�sh , euqarist¸ ton k. StaÔro Hli�dh gia thn episthmonik sunergas�a pou e�qame ìla aut� ta qrìnia, ti parathr sei tou kai th sumpar�stas  tou.Tèlo , euqarist¸ to Panepist mio Patr¸n gia thn upost rixh pou e�qa kat� th di�r-keia th ekpìnhsh th didaktorik  diatrib  mou apì to Ereunhtikì Prìgramma me t�tlo{Jwr�a Diast�sewn kai Kajoliko� Q¸roi} sta pla�sia tou ereunhtikoÔ progr�mmato K.Karajeodwr . Jan�sh Megar�th P�tra, 2010

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Basiko� sumbolismo�; : To kenì sÔnoloA [ B : H ènwsh twn sunìlwn A kai BA \ B : H tom  twn sunìlwn A kai BA� B : To ginìmeno twn sunìlwn A kai BA nB : H diafor� tou sunìlou A apì to B�(X) : H topolog�a tou q¸rou XIntX(M) : To eswterikì tou M sto q¸ro XClX(M) : To per�blhma tou M sto q¸ro XBdX(M) : To sÔnoro tou M sto q¸ro Xind : H mikr  epagwgik  di�stashInd : H meg�lh epagwgik  di�stashdim : H di�stash th kalÔyew O : H kl�sh twn diataktik¸n arijm¸n(+) : To �jroisma tou Hessenberg! : O pr¸to �peiro plhj�rijmo P(X) : To dunamosÔnolo tou sunìlou XjXj : O plhj�rijmo tou sunìlou Xw(X) : To b�ro tou q¸rou X��2�X� : EleÔjero �jroisma twn q¸rwn X�; � 2 �Q�2�X� : Ginìmeno (Ty hono�) twn q¸rwn X�; � 2 �

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PerieqìmenaEisagwg  9MEROS AKef�laio 1Basikè ènnoie : kajoliko� q¸roi kai jewr�a diast�sewn 191.1 To prìblhma kajolikìthta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2 Jewr�a diast�sewn (Genik  Jewr�a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Kef�laio 2Kataskeu  Periektik¸n Q¸rwn 292.1 Prokatarktik� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Shmademènoi q¸roi kai prìtupe sqèsei isodunam�a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3 Oi Periektiko� Q¸roi T(M;R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .352.4 Idi�zonta uposÔnola twn Periektik¸n Q¸rwn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Kef�laio 3Koresmène kl�sei 493.1 Koresmène kl�sei q¸rwn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Koresmène kl�sei uposunìlwn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Koresmène kl�sei b�sewn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Koresmène kl�sei p-b�sewn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54MEROS BKef�laio 4Diast�sei tou tÔpou Ind 594.1 Oi diast�sei dm kai Dm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Oi diast�sei dmIK;IBIE;� kai DmIK;IBIE;� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Sqèsei metaxÔ twn dmIK;IBIE;� , DmIK;IBIE;� kai �llwn diast�sewn . . . . . . . . . . . . . . . . . . . . . . .614.4 Jewr mata ajro�smato kai ginomènou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Idiìthta th kajolikìthta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Kef�laio 5Diast�sei -sunart sei jèsew tou tÔpou ind 757

5.1 Basiko� orismo� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Sqèsei metaxÔ twn diast�sewn jèsew tou tÔpou ind kai �llwn diast�sewn . . . . . 785.3 Jewr mata Upoq¸rou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 Jewr mata Ajro�smato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.5 Apeikon�sei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Kef�laio 6Diast�sei -sunart sei jèsew tou tÔpou Ind 936.1 Basiko� orismo� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Sqèsei metaxÔ twn diast�sewn jèsew tou tÔpou Ind kai �llwn diast�sewn . . . . . 966.3 Jewr mata Upoq¸rou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.4 Jewr mata DiaqwrismoÔ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.5 Jewr mata Ajro�smato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.6 Je¸rhma TaÔtish gia ti pos1-ind kai pos1(0)-Ind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.7 Jewr mata Ginomènou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Kef�laio 7Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 1237.1 Basiko� orismo� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2 Kajolik� stoiqe�a gia diast�sei -sunart sei b�sew jèsew tou tÔpou dim . . . .1307.3 Kajolik� stoiqe�a gia diast�sei -sunart sei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliograf�a 137Per�lhyh 143Abstra t 144

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Eisagwg H Jewr�a Diast�sewn (maz� me th jewr�a twn ontinua) e�nai o palaiìtero kl�do th Genik  Topolog�a . H pr¸th shmantik  prìodo sth Jewr�a Diast�sewn ègine apì tou Poin ar�e, Brouwer kai Lebesgue. H kataskeu  tou Peano to 1890 mia suneqoÔ apeikì-nish apì èna tm ma ep� enì tetrag¸nou èdwse aform  gia to prìblhma e�n èna tm makai èna tetr�gwno e�nai omoiìmorfa, kai genikìtera e�n o n-kÔbo In e�nai omoiìmorfo me ton m-kÔbo Im gia n 6= m. To prìblhma autì lÔjhke apì ton Brouwer to 1911. OBrouwer apèdeixe ìti e�n n 6= m, tìte oi In kai Im den e�nai omoiìmorfoi.O Brouwer prosp�jhse na lÔsei to prìblhma or�zonta mia sun�rthsh df pou anti-stoiqe� se k�je q¸ro èna fusikì arijmì, th di�stash tou q¸rou, me ti idiìthte :(1) E�n oi q¸roi X kai Y e�nai omoiìmorfoi, tìte df(X) = df(Y ).(2) df(In) = n.Wstìso o orismì th topologik  di�stash den  tan aplì z thma. 'Etsi sthn ergas�atou Brouwer (1911) den or�zetai kam�a tètoia sun�rthsh.O Lebesgue to 1911 prìteine mia �llh prosèggish sto prìblhma pou  tan h aform gia ton orismì th di�stash th kalÔyew . DiatÔpwse to je¸rhma ìti o n-kÔbo Inmpore� na kalufje� apì mia peperasmènh oikogèneia auja�reta mikr¸n kleist¸n sunìlwnètsi ¸ste h tom  k�je n+ 2 stoiqe�wn th oikogèneia na e�nai ken  kai ìti den mpore� nakalufje� apì mia peperasmènh oikogèneia auja�reta mikr¸n kleist¸n sunìlwn ètsi ¸steh tom  k�je n + 1 stoiqe�wn th oikogèneia na e�nai ken . H apìdeixh tou jewr mato dìjhke apì ton Brouwer to 1913. O Lebesgue apèdeixe to je¸rhma to 1921. O orismì th di�stash th kalÔyew dim sthn kl�sh twn fusik¸n q¸rwn dìjhke apì ton �Ce hto 1933.To 1912 o Poin ar�e prìteine ènan epagwgikì orismì th di�stash sqetizìmeno me thnènnoia tou diaqwrismoÔ. To 1913 o Brouwer basizìmeno sthn idèa tou Poin ar�e ìriseepagwgik� mia sun�rthsh Dg (Dimensionsgrad) pou antistoiqe� se k�je sumpag  metrikìq¸ro èna fusikì arijmì me ti idiìthte :(1) E�n oi q¸roi X kai Y e�nai omoiìmorfoi, tìte Dg(X) = Dg(Y ).(2) Dg(In) = n.O Brouwer den melèthse prosektik� thn kainoÔrgia di�stash kai th qrhsimopo�hse mìnogia na d¸sei mia �llh apìdeixh ìti oi In kai Im den e�nai omoiìmorfoi gia n 6= m.9

H Jewr�a Diast�sewn ègine anex�rthth perioq  th Genik  Topolog�a met� ti er-gas�e twn Menger kai Urysohn. H jewr�a th mikr  epagwgik  di�stash ind gia thnkl�sh twn sumpag¸n metrik¸n q¸rwn diatup¸jhke kai anaptÔqjhke anex�rthta apì tou Urysohn (1922, 1925, 1926) kai Menger (1923, 1924). H jewr�a epekt�jhke gia thn kl�shtwn diaqwr�simwn metrik¸n q¸rwn apì tou Tumarkin (1925, 1926) kai Hurewi z (1927).O orismì th meg�lh epagwgik  di�stash Ind sthn kl�sh twn fusik¸n q¸rwndìjhke apì ton �Ce h to 1932. O orismì autì moi�zei me ton orismì tou Brouwer all� oidÔo orismo� den e�nai isodÔnamoi. Wstìso, gia tou topik� sunektikoÔ pl rei metrikoÔ q¸rou oi orismo� sump�ptoun.S mera oi diast�sei or�zontai gia opoiod pote topologikì q¸ro. H komyìthta th klasik  Jewr�a Diast�sewn g�netai katanoht  apì to gegonì ìti oi tre� diast�sei sump�ptoun sthn kl�sh twn diaqwr�simwn metrik¸n q¸rwn, dhlad ind(X) = Ind(X) = dim(X)gia k�je diaqwr�simo metrikì q¸ro X. Se megalÔtere kl�sei topologik¸n q¸rwn oiind, Ind kai dim diafèroun. Ep�sh , oi diast�sei Ind kai dim sump�ptoun sthn kl�sh twnmetrik¸n q¸rwn, dhlad  Ind(X) = dim(X)gia k�je metrikì q¸ro X.To 1925 o Urysohn èjese to er¸thma e�n mpore� h mikr  epagwgik  di�stash ind naepektaje� pa�rnonta timè sthn kl�sh ìlwn twn diataktik¸n arijm¸n. Telik�, èna tupi-kì orismì th uperpeperasmènh mikr  epagwgik  di�stash dìjhke apì ton Hurewi zto 1928. H uperpeperasmènh meg�lh epagwgik  di�stash prwtoemfan�sthke to 1959 apìton Smirnov.Shmei¸noume ìti ektì apì ti tre� diast�sei ind, Ind kai dim èqoun orisje� kaimelethje� pollè �lle diast�sei -sunart sei . Wstìso oi tre� autè diast�sei parou-si�zoun to megalÔtero endiafèron.'Ena �llo prìblhma pou apasqìlhse tou topolìgou apì ta pr¸ta b mata an�ptuxh th Genik  Topolog�a sti pr¸te dekaet�e tou ai¸na pou pèrase e�nai to prìblhma th Ôparxh   mh kajolik¸n q¸rwn gia di�fore kl�sei topologik¸n q¸rwn (prìblhma kajo-likìthta ). 'Ena topologikì q¸ro T kale�tai kajolikì gia mia kl�sh IP topologik¸nq¸rwn, ìtan o T an kei sthn kl�sh IP kai k�je topologikì q¸ro pou an kei sthn kl�shIP perièqetai topologik� sto q¸ro T . Anafèroume ta parak�tw klasik� apotelèsmata:10

(1) M. Fr�e het (1910) O q¸ro twn rht¸n arijm¸n Q e�nai kajolikì gia thn kl�sh ìlwntwn arijm simwn metrikopoi simwn q¸rwn.(2) W. Sierpi�nski (1921) To sÔnolo tou Cantor C e�nai kajolikì q¸ro gia thn kl�shìlwn twn mhdenodi�statwn diaqwr�simwn metrikopoi simwn q¸rwn.(3) P. Urysohn (1927) Up�rqei kajolikì q¸ro gia thn kl�sh ìlwn twn diaqwr�simwnmetrikopoi simwn q¸rwn.(4) A. Ty hono� (1930) Gia k�je � � ! o kÔbo tou Ty hono� I� e�nai kajolikì q¸ro gia thn kl�sh ìlwn twn q¸rwn Ty hono� me b�ro �.(5) L. Pontrjagin (1931) K�je diaqwr�simo metrikopoi simo q¸ro diast�sew � n em-futeÔetai ston Eukle�deio q¸ro R2n+1 .(6) P. Alexandro� (1936) Gia k�je � � ! o kÔbo tou Alexandro� S� e�nai kajolikì q¸ro gia thn kl�sh ìlwn twn T0-q¸rwn me b�ro �.(7) H. Kowalsky (1957) Gia k�je � � ! o q¸ro J(�)!, ìpou J(�) e�nai o q¸ro << skantzìqoiro >> me � agk�jia, e�nai kajolikì gia thn kl�sh ìlwn twn metrikopoi simwnq¸rwn me b�ro �.Pollè kl�sei topologik¸n q¸rwn pou èqoun kajolikì q¸ro proèrqontai apì thJewr�a Diast�sewn. Idia�tero endiafèron parousi�zei to ex  prìblhma: 'Estw IP miakl�sh topologik¸n q¸rwn, df mia di�stash-sun�rthsh kai � èna fusikì arijmì . Hkl�sh ìlwn twn q¸rwn X pou an koun sthn kl�sh IP me df(X) � � èqei kajolikì q¸ro;E�n h kl�sh aut  èqei kajolikì q¸ro, tìte lème ìti h di�stash df èqei thn idiìthta th kajolikìthta sthn kl�sh IP. Anafèroume ta parak�tw apotelèsmata:(1) Gia k�je fusikì arijmì � kai plhj�rijmo � � ! h kl�sh ìlwn twn T0-q¸rwn X meind(X) � � kai w(X) � � èqei kajolikì q¸ro.(2) Gia k�je fusikì arijmì � kai plhj�rijmo � � ! h kl�sh ìlwn twn kanonik¸n q¸rwnX me ind(X) � � kai w(X) � � èqei kajolikì q¸ro.(3) Gia k�je fusikì arijmì � kai plhj�rijmo � � ! h kl�sh ìlwn twn metrikopoi simwnq¸rwn X me Ind(X) � � kai w(X) � � èqei kajolikì q¸ro.(4) Gia k�je fusikì arijmì � kai plhj�rijmo � � ! h kl�sh ìlwn twn fusik¸n q¸rwn Xme dim(X) � � kai w(X) � � èqei kajolikì q¸ro.(5) Gia k�je fusikì arijmì � kai plhj�rijmo � � ! h kl�sh ìlwn twn Ty hono� q¸rwnX me dim(X) � � kai w(X) � � èqei kajolikì q¸ro.11

H dom  th didaktorik  diatrib  apart�zetai apì dÔo mèrh kai sunolik� apì ept�kef�laia. To pr¸to mèro , dhlad  ta Kef�laia 1, 2 kai 3, perièqoun basikoÔ orismoÔ kai apotelèsmata qr sima sthn an�ptuxh th didaktorik  diatrib  . Eidikìtera,sto pr¸to kef�laio diatup¸netai to prìblhma kajolikìthta kai parousi�zetai h genik Jewr�a Diast�sewn sthn Topolog�a.Sto deÔtero kef�laio d�netai mia mèjodo kataskeu  periektik¸n q¸rwn gia miaauja�reth oikogèneia T0-q¸rwn, ìpw aut  parousi�zetai sto bibl�o [37℄ (S.D. Iliadis,Universal spa es and mappings, North-Holland Mathemati s Studies, 198. Elsevier S ie-n e B.V., Amsterdam, 2005. xvi+559 pp.). H mèjodo aut  e�nai sunolojewrhtik  kaiqrhsimopoie�tai sta Kef�laia 4 kai 7 gia thn kataskeu  Kajolik¸n Q¸rwn.To tr�to kef�laio perièqei orismoÔ kai prot�sei pou perièqontai sto bibl�o [37℄.Eidikìtera, d�netai o orismì th koresmènh kl�sh q¸rwn kai h ènnoia th koresmènh kl�sh q¸rwn pou èqoun mia {dom }. Kl�sei q¸rwn me dom  e�nai oi kl�sei uposunìlwn,oi kl�sei b�sewn kai oi kl�sei p-b�sewn. Gia ti kl�sei autè d�netai h ènnoia toukajolikoÔ stoiqe�ou. Oi koresmène kl�sei èqoun thn idiìthta th kajolikìthta , dhlad se k�je koresmènh kl�sh up�rqei kajolikì stoiqe�o. Wstìso, oi koresmène kl�sei q¸rwn èqoun {k�ti parap�nw} apì thn Ôparxh kajolik¸n q¸rwn. Gia par�deigma, oikoresmène kl�sei èqoun thn idiìthta th tom  , dhlad  h tom  koresmènwn kl�sewne�nai ep�sh mia koresmènh kl�sh, parìlo pou h tom  kl�sewn pou èqoun kajolik� stoiqe�ampore� na mhn èqei kajolikì stoiqe�o.Ta apotelèsmata tou deÔterou mèrou , dhlad  ta apotelèsmata twn Kefala�wn4, 5, 6 kai 7, e�nai prwtìtupa kai èqoun dhmosieuje� se diejn  periodik� (blèpe [23℄,[24℄, [25℄, [26℄ kai [27℄). Sto deÔtero mèro , upojètoume ìti ìloi topologiko� q¸roi e�naiT0-q¸roi me b�ro � � , ìpou � e�nai èna stajerì �peiro plhj�rijmo .Sthn ergas�a [56℄ or�sjhkan dÔo diast�sei dm kai Dm sthn kl�sh ìlwn twn Haus-dor� q¸rwn. H di�stash Dm den èqei thn idiìthta th kajolikìthta sthn kl�sh ìlwntwn diaqwr�simwn metrikopoi simwn q¸rwn epeid  h oikogèneia ìlwn twn diaqwr�simwn me-trikopoi simwn q¸rwn X me Dm(X) � 0 sump�ptei me thn oikogèneia ìlwn twn totally di-s onne ted q¸rwn sthn opo�a den up�rqoun kajolik� stoiqe�a (blèpe [65℄). Sto tètartokef�laio tropopoioÔntai oi diast�sei dm kai Dm me skopì na orisjoÔn nèe diast�-sei pou èqoun thn idiìthta th kajolikìthta . Autè oi nèe diast�sei sumbol�zontai medmIK;IBIE;� kai DmIK;IBIE;� , ìpou IE e�nai mia kl�sh q¸rwn, IK e�nai mia kl�sh uposunìlwn kai IBe�nai mia kl�sh b�sewn.Analutikìtera, sthn pr¸th kai sth deÔterh par�grafo d�nontai oi orismo� twn diast�-sewn dm, Dm, dmIK;IBIE;� kai DmIK;IBIE;� . 12

Sthn tr�th par�grafo d�nontai sqèsei metaxÔ twn diast�sewn dm, Dm, dmIK;IBIE;� kaiDmIK;IBIE;� kai sugkr�nontai me �lle gnwstè diast�sei . Eidikìtera, apodeiknÔetai ìti e�no q¸ro X e�nai sumpag  Hausdor� me DmIK;IB! (X) 6= 1, tìte oi parak�tw sunj ke e�nai isodÔname : (1) DmIK;IB! (X) = 0, (2) dmIK;IB! (X) = 0 kai (3) Ind(X) = 0.Sthn tètarth par�grafo diatup¸nontai kai apodeiknÔontai jewr mata ajro�smato kaiginomènou.Tèlo , sthn pèmpth par�grafo apodeiknÔetai ìti e�n oi kl�sei IK; IB kai IE e�naikoresmène , tìte gia mia dosmènh koresmènh kl�sh IP q¸rwn kai gia èna fusikì arijmì� h kl�sh ìlwn twn q¸rwn X pou an koun sthn kl�sh IP me dmIK;IBIE;� (X) � � kai hkl�sh ìlwn twn q¸rwn X pou an koun sthn kl�sh IP me DmIK;IBIE;� (X) � � èqoun kajolik�stoiqe�a. ApodeiknÔetai ìti e�n IP e�nai mia apì ti parak�tw kl�sei :(1) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn me b�ro � � ,(2) h kl�sh ìlwn twn (pl rw ) kanonik¸n ountable-dimensional q¸rwn me b�ro � � ,(3) h kl�sh ìlwn twn (pl rw ) kanonik¸n strongly ountable-dimensional q¸rwn me b�ro � � ,(4) h kl�sh ìlwn twn (pl rw ) kanonik¸n lo ally �nite-dimensional q¸rwn me b�ro � �kai(5) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn X me w(X) � � kai ind(X) � � 2 �+,ìpou �+ e�nai o mikrìtero plhj�rijmo pou e�nai megalÔtero apì to � ,tìte gia k�je � 2 ! sti kl�sei IP(dmIK;IBIE;� � �) \ IP kai IP(DmIK;IBIE;� � �) \ IP up�rqounkajolik� stoiqe�a.Sto bibl�o [37℄ or�sjhkan diast�sei -sunart sei jèsew tou tÔpou ind. Oi diast�sei autè èqoun ped�o orismoÔ thn kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna uposÔnoloenì q¸rou X, kai sumbol�zontai me pi-ind, posi-ind kai psi-ind, i = 0; 1. Oi parap�nwdiast�sei melet jhkan mìno ìson afor� thn idiìthta th kajolikìthta , dhlad  e�n dfe�nai mia apì ti parap�nw sunart sei kai � 2 �+, ìpou �+ e�nai o mikrìtero plhj�-rijmo pou e�nai megalÔtero apì to � , tìte sthn kl�sh IP ìlwn twn zeug¸n (QX ; X),ìpou QX e�nai èna uposÔnolo enì q¸rou X me df(QX ; X) � �, up�rqei kajolikì stoi-qe�o. ('Ena stoiqe�o (QT ; T ) th IP kale�tai kajolikì sthn IP e�n gia k�je (QX ; X) 2 IPup�rqei mia topologik  emfÔteush iXT : X ! T tètoia ¸ste iXT (QX) � QT .) Sto pèmptokef�laio d�nontai sqèsei metaxÔ twn diast�sewn-sunart sewn jèsew tou tÔpou ind kaiapodeiknÔontai basikè idiìthte th Jewr�a Diast�sewn gia ti sunart sei autè .Analutikìtera, sthn pr¸th par�grafo d�nontai oi orismo� twn diast�sewn-sunart sewnjèsew pi-ind, posi-ind kai psi-ind, i = 0; 1.13

Sth deÔterh par�grafo d�nontai sqèsei metaxÔ twn diast�sewn-sunart sewn jèsew pi-ind, posi-ind kai psi-ind, i = 0; 1, kai sugkr�nontai me �lle gnwstè diast�sei . Apo-deiknÔetai ìti gia k�je uposÔnolo Q enì q¸rou X isqÔeiind(Q) � pi-ind(Q;X) � posi-ind(Q;X) � psi-ind(Q;X); i = 0; 1:ApodeiknÔetai ìti e�n o q¸ro X e�nai klhronomik� fusikì (dhlad  k�je upìqwro touX e�nai fusikì ) kai Q � X, tìteind(Q) = p1-ind(Q;X) = pos1-ind(Q;X):Ep�sh , d�nontai parade�gmata pou de�qnoun ìti sthn kl�sh ìlwn twn T0-q¸rwn oi para-p�nw anisìthte den mporoÔn na antikatastajoÔn me isìthte .Sthn tr�th par�grafo diatup¸nontai kai apodeiknÔontai jewr mata upoq¸rou. Eidi-kìtera, apodeiknÔetai ìti e�n Y e�nai èna puknì uposÔnolo enì q¸rou X kai Q � Y ,tìte p1-ind(Q; Y ) = p1-ind(Q;X)kai pos1-ind(Q; Y ) = pos1-ind(Q;X):Sthn tètarth par�grafo diatup¸nontai kai apodeiknÔontai jewr mata ajro�smato .Tèlo , sthn pèmpth par�grafo d�nontai sqèsei metaxÔ twn diast�sewn tou ped�ouorismoÔ kai tou ped�ou tim¸n mia suneqoÔ apeikìnish .Sthn ergas�a [40℄ (blèpe ep�sh [30℄) or�sjhke mia di�stash-sun�rthsh jèsew toutÔpou Ind. Sto èkto kef�laio or�zontai nèe diast�sei -sunart sei jèsew tou tÔpouInd kai apodeiknÔontai basikè idiìthte th Jewr�a Diast�sewn gia ti sunart sei autè . Oi diast�sei autè èqoun ped�o orismoÔ thn kl�sh ìlwn twn zeug¸n (Q;X),ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai sumbol�zontai me pi(j)-ind kai posi(j)-ind,i = 0; 1, j = 0; 1.Analutikìtera, sthn pr¸th par�grafo d�nontai oi orismo� twn diast�sewn sunart -sewn jèsew pi(j)-ind kai posi(j)-ind, i = 0; 1, j = 0; 1.Sth deÔterh par�grafo d�nontai sqèsei metaxÔ twn diast�sewn-sunart sewn jèsew pi(j)-ind kai posi(j)-ind, i = 0; 1, j = 0; 1, kai sugkr�nontai me �lle gnwstè diast�sei .ApodeiknÔetai ìti e�n o q¸ro X e�nai klhronomik� fusikì (dhlad  k�je upìqwro touX e�nai fusikì ) kai Q � X, tìtep1(1)-Ind(Q;X) = pos1(1)-Ind(Q;X) = Ind(Q):14

Ep�sh , d�nontai parade�gmata pou de�qnoun ìti oi sunart sei pi(j)-ind kai posi(j)-ind,i = 0; 1, j = 0; 1, e�nai diaforetikè .Sthn tr�th par�grafo diatup¸nontai kai apodeiknÔontai jewr mata upoq¸rou.Sthn tètarth par�grafo diatup¸nontai kai apodeiknÔontai jewr mata diaqwrismoÔ.Sthn pèmpth par�grafo diatup¸nontai kai apodeiknÔontai jewr mata ajro�smato .Sthn èkth par�grafo d�netai èna je¸rhma taÔtish gia ti diast�sei pos1-ind kaipos1(0)-Ind.Tèlo , sthn èbdomh par�grafo diatup¸nontai kai apodeiknÔontai jewr mata ginomènou.Sto bibl�o [37℄ or�sjhkan diast�sei -sunart sei b�sew tou tÔpou ind, Ind kai dim.Oi diast�sei -sunart sei autè melet jhkan mìno w pro thn idiìthta th kajolikìth-ta . Sto èbdomo kef�laio or�zontai diast�sei -sunart sei b�sew jèsew tou tÔpoudim kai apodeiknÔetai h idiìthta th kajolikìthta gia ti sunart sei autè . Oi dia-st�sei autè èqoun ped�o orismoÔ thn kl�sh ìlwn twn tri�dwn (Q;B;X), ìpou Q e�naièna uposÔnolo enì q¸rou X kai B e�nai mia oikogèneia apì anoikt� uposÔnola tou X(sumperilambanomènwn twn X kai ;) tètoia ¸ste to sÔnolo fQ \ U : U 2 Bg na e�nai miab�sh tou upoq¸rou Q, kai sumbol�zontai me b-p0-dimIF, b-p1-dimIF kai b-p-dimIF, ìpou IFe�nai mia kl�sh uposunìlwn.Analutikìtera, sthn pr¸th par�grafo d�nontai sqèsei metaxÔ twn sunart sewn b�-sew jèsew b-p0-dimIF, b-p1-dimIF kai b-p-dimIF kai sugkr�nontai me �lle gnwstè diast�sei . Ep�sh , d�nontai parade�gmata pou de�qnoun ìti oi sunart sei b-p0-dimIF,b-p1-dimIF kai b-p-dimIF e�nai diaforetikè .Sth deÔterh par�grafo apodeiknÔetai ìti e�n df e�nai mia apì ti diast�sei -sunart sei b�sew jèsew : b-p0-dimIF kai b-p1-dimIF kai b-p-dimIF kai e�n h kl�sh IF e�nai koresmè-nh, pl rh kai ikanopoie� ti Sunj ke Peperasmènh 'Enwsh kai KenoÔ Uposunìlou,tìte gia k�je n 2 f�1g [ ! h kl�sh IP(df � n) èqei kajolik� stoiqe�a ('Ena stoiqe�o(QT ; BT ; T ) th IP kale�tai kajolikì sthn IP e�n gia k�je (QZ ; BZ ; Z) 2 IP up�rqei miatopologik  emfÔteush e : Z ! T tètoia ¸ste e(QZ) � QT kai BZ = fe�1(V ) : V 2 BTg.)Tèlo , sthn tr�th par�grafo e�n df e�nai mia apì ti diast�sei -sunart sei b�sew jèsew : b-p-dimIF, b-p0-dimIF kai b-p1-dimIF kai ID e�nai mia kl�sh apì tri�de (Q;B;X),ìpou Q e�nai èna uposÔnolo enì q¸rou X kai B e�nai mia oikogèneia apì anoikt� uposÔno-la tou X (sumperilambanomènwn twn X kai ;) tètoia ¸ste to sÔnolo fQ\U : U 2 Bg nae�nai mia b�sh tou upoq¸rou Q, tìte or�zetai mia nèa di�stash-sun�rthsh ID-df , me ped�oorismoÔ thn kl�sh ìlwn twn q¸rwn kai apodeiknÔetai ìti e�n h kl�sh ID e�nai koresmènhkai h kl�sh IF e�nai koresmènh, pl rh kai ikanopoie� ti Sunj ke Peperasmènh 'Enwsh kai KenoÔ Uposunìlou, tìte gia k�je n 2 f�1g[! h kl�sh IP(ID-df � n) èqei kajolik�15

stoiqe�a. ApodeiknÔetai ìti e�n IF e�nai h kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�naièna anoiktì uposÔnolo enì q¸rou X, ID e�nai h kl�sh ìlwn twn tri�dwn (Q;B;X), ìpouQ e�nai èna uposÔnolo enì q¸rou X kai B e�nai mia oikogèneia apì anoikt� uposÔnolatou X (sumperilambanomènwn twn X kai ;) tètoia ¸ste to sÔnolo fQ \ U : U 2 Bg nae�nai mia b�sh tou upoq¸rou Q, kai IP mia apì ti parak�tw kl�sei :(1) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn me b�ro � � ,(2) h kl�sh ìlwn twn (pl rw ) kanonik¸n ountable-dimensional q¸rwn me b�ro � � ,(3) h kl�sh ìlwn twn (pl rw ) kanonik¸n strongly ountable-dimensional q¸rwn me b�ro � � ,(4) h kl�sh ìlwn twn (pl rw ) kanonik¸n lo ally �nite-dimensional q¸rwn me b�ro � �kai(5) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn X me w(X) � � kai ind(X) � � 2 �+,ìpou �+ e�nai o mikrìtero plhj�rijmo pou e�nai megalÔtero apì to � ,tìte gia k�je n 2 ! sthn kl�sh IP(ID-df � n) \ IP up�rqoun kajolik� stoiqe�a.

16

MEROS A

Kef�laio 1Basikè ènnoie : kajoliko� q¸roikai jewr�a diast�sewnSto kef�laio autì diatup¸netai to prìblhma kajolikìthta pou èqoume sth Genik Topolog�a kai parousi�zetai h genik  Jewr�a Diast�sewn.1.1 To prìblhma kajolikìthta Sthn par�grafo aut  d�netai o orismì tou kajolikoÔ q¸rou mia kl�sh topologik¸nq¸rwn kai diatup¸netai to prìblhma kajolikìthta pou èqoume sth Genik  Topolog�a.1.1.1 Orismì . 'Estw X topologikì q¸ro . Mia idiìthta P tou q¸rou X kale�tai to-pologik , e�n k�je topologikì q¸ro Y omoiìmorfo me ton X èqei ep�sh thn idiìthtaaut .1.1.2 Orismì . 'Estw IP kl�sh topologik¸n q¸rwn. H IP kale�tai topologik   topologik¸ kleist , e�n k�je topologikì q¸ro Y pou e�nai omoiìmorfo me ènaq¸ro X th kl�sh IP an kei sthn kl�sh IP. H ken  kl�sh q¸rwn jewre�tai topologik .1.1.3 Orismì . 'Estw IP kl�sh topologik¸n q¸rwn. 'Ena topologikì q¸ro T kale�taikajolikì (universal spa e) gia thn kl�sh IP, ìtan:(1) T 2 IP.(2) Gia k�je X 2 IP, up�rqei topologik  emfÔteush eX : X ! T .Sun jw , h kl�sh IP or�zetai w mia kl�sh topologik¸n q¸rwn pou èqei mia sugkekri-mènh topologik  idiìthta P. Se aut  thn per�ptwsh ja lème ìti o q¸ro T e�nai kajolikì gia ìlou tou topologikoÔ q¸rou pou èqoun aut  thn idiìthta, e�n o T èqei thn idiì-thta P kai k�je topologikì q¸ro X pou èqei thn idiìthta P e�nai omoiìmorfo me ènanupìqwro tou T . 19

20 Kef�laio 11.1.4 Orismì . 'Estw IP kl�sh topologik¸n q¸rwn. 'Ena topologikì q¸ro T kale�taiperiektikì ( ontaining spa e) gia thn kl�sh IP, e�n gia k�je q¸ro X pou an kei sthnkl�sh IP up�rqei èna omoiomorfismì tou X me ènan upìqwro tou T .1.1.5 Parat rhsh. K�je oikogèneia topologik¸n q¸rwn èqei periektikì q¸ro. Pr�g-mati, èstw fX� : � 2 �g m�a oikogèneia topologik¸n q¸rwn. Gia k�je � 2 � jewroÔme toq¸ro X 0� = X� � f�g me thn topolog�a�(X 0�) = fU � f�g : U 2 �(X�)g:Profan¸ X 0� \X 0�0 = ; gia � 6= �0 kai o q¸ro X 0� e�nai omoiìmorfo me to q¸ro X� giak�je � 2 �.JewroÔme to sÔnolo T = [fX 0� : � 2 �g kai or�zoume mia topolog�a � ep� tou T w ex  : � = fU � T : gia k�je � 2 �; U \X 0� 2 �(X 0�)g:O topologikì q¸ro (T; �) sumbol�zetai me ��2�X� kai kale�tai eleÔjero �jroisma(free sum) twn q¸rwn X�; � 2 �.E�nai fanerì ìti gia k�je � 2 �, h apeikìnish i� : X� ! ��2�X� me tÔpo i�(x) = (x; �)e�nai m�a emfÔteush tou X� sto ��2�X�.Apì tou parap�nw orismoÔ prokÔptei �mesa to parak�tw prìblhma.Prìblhma. 'Estw IP kl�sh topologik¸n q¸rwn. Up�rqei kajolikì q¸ro gia thnkl�sh IP ;To parap�nw prìblhma onom�zetai prìblhma kajolikìthta (universality problem)gia thn kl�sh IP.Shmei¸noume ìti ta jewr mata pou anafèrontai sthn Ôparxh kajolik¸n q¸rwn e�naipolÔ endiafèronta kai qr sima. Gia par�deigma, ma epitrèpoun na anag�goume th melèthmia kl�sh topologik¸n q¸rwn pou èqoun mia sugkekrimènh idiìthta, sth melèth twnupoq¸rwn enì stajeroÔ topologikoÔ q¸rou.A doÔme t¸ra k�poie parathr sei sqetikè me to prìblhma kajolikìthta .1.1.6 Parat rhsh. 'Estw IP kl�sh topologik¸n q¸rwn pou èqei kajolikì q¸ro. E�nIP1 kai IP2 e�nai dÔo kl�sei topologik¸n q¸rwn tètoie ¸ste IP1 � IP � IP2, tìte denprokÔptei ìti oi IP1 kai IP2 èqoun kajolikì q¸ro.

Basikè ènnoie : kajoliko� q¸roi kai jewr�a diast�sewn 21H parap�nw parat rhsh kajist� to prìblhma twn kajolik¸n q¸rwn akìmh pio endia-fèron.1.1.7 Parat rhsh. 'Estw IP kl�sh topologik¸n q¸rwn pou èqei kajolikì q¸ro T .Tìte, gia k�je topologikì q¸ro X pou an kei sthn kl�sh IP isqÔoun:(1) jXj � jT j kai(2) w(X) � w(T ).Pr�gmati, ef> ìson o T e�nai kajolikì , up�rqei emfÔteush eX : X ! T . Epomènw jXj � jT j. 'Estw t¸ra BT = fV� : � 2 �g b�sh tou T me j�j = w(T ) kai Y = eX (X).Upojètoume ìti U 2 �(X). Tìte, eX (U) 2 �(Y ) kai �ra up�rqei V 2 �(T ) tètoio ¸steeX (U) = V \ Y . Ef> ìson h BT e�nai b�sh tou T kai V 2 �(T ), up�rqei K � � tètoio¸ste V = [�2KV�. 'EqoumeeX (U) = ([�2KV�) \ Y = [�2K(V� \ Y )  U = e�1X � [�2K (V� \ Y )� = [�2Ke�1X (V� \ Y ) = [�2Ke�1X (V�):'Ara, to sÔnolo BX � fe�1X (V�) : � 2 �ge�nai b�sh tou X kai epiplèonw(X) � jBX j = jBT j = w(T ):Sunep¸ , gia na èqei ènnoia na anazht soume kajolikì q¸ro se mia kl�sh IP, prèpeigia ton plhj�rijmo jXj kai gia to b�ro w(X) twn q¸rwn X 2 IP na èqoume èna �nwfr�gma. Se ant�jeth per�ptwsh, h kl�sh IP den èqei kajolikì q¸ro. Gia par�deigma, hkl�sh T ìlwn twn tetrimmènwn topologik¸n q¸rwn den èqei kajolikì q¸ro. Profan¸ isqÔei w(X) = 1 gia k�je X 2 T, all� oi plhj�rijmoi jXj den e�nai fragmènoi. To �dioisqÔei kai gia thn kl�sh ìlwn twn diakritik¸n q¸rwn, afoÔ sthn per�ptwsh aut  oÔte tob�ro twn q¸rwn e�nai fragmèno.1.1.8 Prìtash. 'Estw IP kl�sh T0-q¸rwn me w(X) � � gia k�je X 2 IP, gia k�poioplhj�rijmo �. Tìte, jXj � 2� gia k�je X 2 IP.Apìdeixh. 'Estw X 2 IP kai B = fV� : � 2 �g b�sh tou X me jBj � �. Tìte,X = [fV� : � 2 �g. JewroÔme thn apeikìnish � : X ! 2B pou or�zetai w ex  : 'Estw

22 Kef�laio 1x 2 X. Tìte, �(x) = f , ìpouf : B ! 2 me f(V�) = 8<:1; e�n x 2 V�;0; diaforetik�:ApodeiknÔoume ìti h � e�nai 1-1. Pr�gmati, èstw x; y 2 X me �(x) = f kai �(y) = g.Upojètoume ìti �(x) = �(y). Ja de�xoume ìti kai x = y. 'Estw ant�jeta ìti x 6= y. Jakatal xoume se �topo. Epeid  o X e�nai T0-q¸ro , up�rqei anoiktì sÔnolo U tou X pouperièqei, gia par�deigma, to x kai den perièqei to y. 'Ara, up�rqei k�poio �0 2 � tètoio¸ste x 2 V�0 � U kai y =2 V�0. Autì ìmw shma�nei ìti f(V�0) = 1 en¸ g(V�0) = 0. 'Atopo,afoÔ e�qame upojèsei ìti f = g. Sunep¸ , jXj � 2�. �1.1.9 Parat rhsh. Mia kl�sh topologik¸n q¸rwn mpore� na èqei diaforetikoÔ (mhomoiìmorfou ) kajolikoÔ q¸rou .1.1.10 Gnwst� Apotelèsmata.(1) O q¸ro twn pragmatik¸n arijm¸n R e�nai periektikì gia thn kl�sh ìlwn twn pepe-rasmènwn metrikopoi simwn q¸rwn.(2) O q¸ro twn rht¸n arijm¸n Q e�nai kajolikì gia thn kl�sh ìlwn twn arijm simwnmetrikopoi simwn q¸rwn.(3) H kl�sh ìlwn twn topologik¸n q¸rwn me n shme�a (n � 2) den èqei kajolikì q¸ro.(4) O q¸ro C([0; 1℄) twn suneq¸n pragmatik¸n sunart sewn pou or�zontai sto [0; 1℄ me thmetrik  th omoiìmorfh sÔgklish e�nai kajolikì gia thn kl�sh ìlwn twn diaqwr�simwnmetrikopoi simwn q¸rwn.To prìblhma kajolikìthta gia di�fore kl�sei topologik¸n q¸rwn apasqìlhse tou topolìgou apì ta pr¸ta b mata an�ptuxh th Genik  Topolog�a sti pr¸te dekaet�e tou ai¸na pou pèrase. Sthn arq  den up rqe genik  mèjodo kataskeu  kajolik¸n q¸rwnkai oi kataskeuè tètoiwn q¸rwn bas�zontan kur�w sth dia�sjhsh. Wstìso shmantikìrìlo sthn kataskeu  periektik¸n kai kajolik¸n q¸rwn pa�zoun ta ginìmena topologik¸nq¸rwn. Parak�tw diatup¸netai èna jèwrhma emfÔteush se ginìmena topologik¸n q¸rwn.'Estw X èna topologikì q¸ro , fY� : � 2 �g oikogèneia topologik¸n q¸rwn kaiff� : � 2 �g oikogèneia apeikon�sewn, ìpou f� : X ! Y� gia k�je � 2 �.1.1.11 Orismì . H apeikìnish f : X ! Q�2� Y� me tÔpo f(x) = ff�(x)g�2� kale�taidiag¸nio apeikìnish twn apeikon�sewn f�, � 2 � (diagonal of the mappings f�, � 2 �)kai sumbol�zetai me 4�2�f�.

Basikè ènnoie : kajoliko� q¸roi kai jewr�a diast�sewn 231.1.12 Orismì . (1) H oikogèneia ff� : � 2 �g diaqwr�zei ta shme�a tou X (separatespoints of the spa e X), e�n gia k�je x; y 2 X me x 6= y, up�rqei �0 2 � tètoio ¸stef�0(x) 6= f�0(y).(2) H oikogèneia ff� : � 2 �g diaqwr�zei ta shme�a kai ta kleist� uposÔnola touX (separates points from losed sets), e�n gia k�je kleistì sÔnolo F tou X kai gia k�jex 2 X me x =2 F , up�rqei �0 2 � tètoio ¸ste f�0(x) =2 Cl(f�0(F )).1.1.13 Prìtash. (1) E�n h oikogèneia ff� : � 2 �g diaqwr�zei ta shme�a tou X, tìte hdiag¸nio apeikìnish 4�2�f� e�nai 1-1.(2) E�n h oikogèneia ff� : � 2 �g apotele�tai apì suneqe� sunart sei , diaqwr�zei tashme�a tou X kai diaqwr�zei ta shme�a kai ta kleist� uposÔnola tou X, tìte h diag¸nio apeikìnish 4�2�f� e�nai emfÔteush.1.1.14 Parat rhsh. E�n o q¸ro X e�nai T0 kai h oikogèneia ff� : � 2 �g diaqwr�zeita shme�a kai ta kleist� uposÔnola tou X, tìte h ff� : � 2 �g diaqwr�zei kai ta shme�atou X.Ta parak�tw apotelèsmata e�nai qarakthristik� parade�gmata qr sh th Prìtash 1.1.13.1.1.15 Gnwst� Apotelèsmata.(1) 'Estw o topologikì q¸ro (E; �), ìpou E = f0; 1; 2g kai � = f;; f0g; Eg. Giak�je � � ! o q¸ro E� e�nai kajolikì gia thn kl�sh ìlwn twn q¸rwn me b�ro � kaiplhj�rijmo � 2�.(2) 'Estw S o q¸ro tou Sierpi�nski, dhlad  to sÔnolo S = f0; 1g me thn topolog�af;; f0g; f0; 1gg. Gia k�je � � ! o q¸ro S� e�nai kajolikì gia thn kl�sh ìlwn twnT0-q¸rwn me b�ro �.(3) 'Estw o q¸ro I = [0; 1℄ me th sun jh metrik . O q¸ro I� e�nai kajolikì gia thnkl�sh ìlwn twn q¸rwn Ty hono� me b�ro �.(4) Gia k�je � � ! o q¸ro J(�)!, ìpou J(�) e�nai o q¸ro skantzìqoiro me � agk�jia,e�nai kajolikì gia thn kl�sh ìlwn twn metrikopoi simwn q¸rwn me b�ro �.Me th melèth pio genik¸n kl�sewn topologik¸n q¸rwn, proerqìmene kur�w apì thJewr�a Diast�sewn, proèkuyan merikè mèjodoi kataskeu  kajolik¸n q¸rwn. Oi mèjo-doi pou qrhsimopoioÔn jewr mata paragontopo�hsh (fa torization theorems) fa�nontai nae�nai oi pio shmantikè .

24 Kef�laio 11.2 Jewr�a diast�sewn (Genik  Jewr�a)Se ì,ti akolouje� me O sumbol�zoume thn kl�sh twn diataktik¸n arijm¸n kai me !ton pr¸to �peiro plhj�rijmo. Ep�sh , jewroÔme dÔo sÔmbola <<�1>> kai <<1>> tètoia ¸ste�1 < � <1 gia k�je � 2 O kai �1(+)� = �(+)(�1) = �, 1(+)� = �(+)1 =1 giak�je � 2 O [ f�1;1g.Mia <<di�stash-sun�rthsh>> (dimension-like fun tion)   apl� <<di�stash>> e�nai mia su-n�rthsh df me ped�o orismoÔ thn kl�sh ìlwn twn q¸rwn kai ped�o tim¸n to sÔnolo! [ f�1;1g   thn kl�sh O [ f�1;1g, me ti idiìthte :(1) E�n oi q¸roi X kai Y e�nai omoiìmorfoi, tìte df(X) = df(Y ).(2) df(Rn) = n.H Jewr�a Diast�sewn melet�ei ti idiìthte aut¸n twn sunart sewn. Oi kuriìere idiì-thte twn diast�sewn-sunart sewn e�nai oi ex  :1. Jewr mata upoq¸rou. 'Estw M upìqwro enì topologikoÔ q¸rou X. K�tw apìpoie sunj ke isqÔei h anisìthta df(M) � df(X) ;2. Jewr mata ajro�smato . 'Estw fF� : � 2 �g k�lumma enì topologikoÔ q¸rou Xètsi ¸ste df(F�) � n, ìpou n 2 !, gia k�je � 2 �. K�tw apì poie sunj ke isqÔei haniisìthta df(X) � n ;3. Anisìthta tou Urysohn. K�tw apì poie sunj ke isqÔei h anisìthtadf(A [B) � df(A) + df(B) + 1 ;4. Jewr mata ginomènou. K�tw apì poie sunj ke isqÔei h anisìthtadf(X � Y ) � df(X) + df(Y ) ;5. Idiìthta th kajolikìthta . Gia poiou diataktikoÔ arijmoÔ � h kl�sh ìlwn twnq¸rwn X me df(X) � � èqei kajolikì q¸ro ;H Jewr�a Diast�sewn exet�zei ep�sh ti sqèsei metaxÔ diaforetik¸n diast�sewn.'Ena shmantikì prìblhma e�nai h eÔresh sunjhk¸n k�tw apì ti opo�e k�poie diast�sei sump�ptoun.Sthn par�grafo aut  or�zontai oi tre� kuriìtere diast�sei : ind, Ind kai dim kaidiatup¸nontai oi basikè idiìthte aut¸n.1.2.1 Orismì . 'Estw A kai B dÔo xèna uposÔnola enì q¸rou X. Ja lème ìti ènauposÔnolo L tou X diaqwr�zei (separates) ta sÔnola A kai B   ìti e�nai mia diamèrish

Basikè ènnoie : kajoliko� q¸roi kai jewr�a diast�sewn 25(partition) metaxÔ twn A kai B e�n up�rqoun dÔo anoikt� uposÔnola U kai W tou X ètsi¸ste: (1) A � U , B � W , (2) U \W = ; kai (3) X n L = U [W .H Mikr  Epagwgik  Di�stash1.2.2 Orismì . JewroÔme th sun�rthsh ind me ped�o orismoÔ thn kl�sh ìlwn twn q¸rwnkai ped�o tim¸n to sÔnolo ! [ f�1;1g pou ikanopoie� ta parak�tw axi¸mata:A1) ind(X) = �1 e�n kai mìno e�n X = ;.A2) ind(X) � n, ìpou n 2 !, e�n up�rqei b�sh B tou X ètsi ¸ste ind(Bd(U)) < ngia k�je U 2 B.H ind kale�tai mikr  epagwgik  di�stash (small indu tive dimension)   di�stash twnMenger-Urysohn.1.2.3 Parat rhsh. Sun jw h di�stash ind or�zetai gia kanonikoÔ q¸rou   gia k�poiapio periorismènh kl�sh q¸rwn. ApodeiknÔetai ìti èna kanonikì q¸ro X ikanopoie� thnanisìthta ind(X) � n, ìpou n 2 !, e�n kai mìno e�n gia k�je x 2 X kai k�je kleistìuposÔnolo F tou X me x =2 F up�rqei diamèrish L metaxÔ twn fxg kai F tètoia ¸steind(L) < n.1.2.4 Parat rhsh. 'Estw IP mia topologik  kl�sh. E�n ston Orismì 1.2.2 antikata-st soume to ax�wma A1) me to ax�wma:A10) ind(X) = �1 e�n kai mìno e�n X 2 IP,tìte pa�rnoume th mikr  epagwgik  di�stash modulo IP.1.2.5 Je¸rhma. Gia k�je upìqwro M enì q¸rou X, ind(M) � ind(X).1.2.6 Je¸rhma. 'Estw X klhronomik� fusikì q¸ro . E�n X = X1 [X2, tìteind(X) � ind(X1) + ind(X2) + 1:H Meg�lh Epagwgik  Di�stash1.2.7 Orismì . JewroÔme th sun�rthsh Ind me ped�o orismoÔ thn kl�sh ìlwn twn q¸rwnkai ped�o tim¸n to sÔnolo ! [ f�1;1g pou ikanopoie� ta parak�tw axi¸mata:A1) Ind(X) = �1 e�n kai mìno e�n X = ;.A2) Ind(X) � n, ìpou n 2 !, e�n gia k�je kleistì sÔnolo F tou X kai gia k�jeanoiktì sÔnolo V tou X me F � V up�rqei anoiktì sÔnolo U tou X tètoio

26 Kef�laio 1¸ste F � U � V kai Ind(Bd(U)) < n.H Ind kale�tai meg�lh epagwgik  di�stash (large indu tive dimension)   di�stashtwn Brouwer-�Ce h.1.2.8 Parat rhsh. Sun jw h di�stash Ind or�zetai gia fusikoÔ q¸rou   gia k�poiapio periorismènh kl�sh q¸rwn. ApodeiknÔetai ìti èna fusikì q¸ro X ikanopoie� thnanisìthta Ind(X) � n, ìpou n 2 !, e�n kai mìno e�n gia k�je zeÔgo A;B xènwn kleist¸nuposunìlwn tou X up�rqei diamèrish L metaxÔ twn A kai B tètoia ¸ste Ind(L) < n.1.2.9 Parat rhsh. 'Estw IP mia topologik  kl�sh. E�n ston Orismì 1.2.7 antikata-st soume to ax�wma A1) me to ax�wma:A10) Ind(X) = �1 e�n kai mìno e�n X 2 IP,tìte pa�rnoume th meg�lh epagwgik  di�stash modulo IP.1.2.10 Je¸rhma. Gia k�je kleistì upìqwro M enì topologikoÔ q¸rou X,Ind(M) � Ind(X):1.2.11 Je¸rhma. 'Estw X klhronomik� fusikì q¸ro . E�n X = X1 [X2, tìteInd(X) � Ind(X1) + Ind(X2) + 1:1.2.12 Je¸rhma. 'Estw X; Y metrikopoi simoi q¸roi me X [ Y 6= ;. Tìte,Ind(X � Y ) � Ind(X) + Ind(Y ):1.2.13 Je¸rhma. 'Estw X fusikì q¸ro . E�n up�rqei akolouj�a F1; F2; : : : ; kleist¸nuposunìlwn tou X ètsi ¸ste Ind(Fi) � 0 gia i = 1; 2; : : : ; kai X = [1i=1Fi, tìteInd(X) � 0:1.2.14 Orismì . Mia oikogèneia uposunìlwn enì topologikoÔ q¸rou X kale�taik�lumma ( over) tou X e�n X = [A2 A. To k�lumma kale�tai anoiktì (open) e�n ìlata stoiqe�a tou e�nai anoikt� uposÔnola tou X.1.2.15 Orismì . 'Estw X topologikì q¸ro kai 1; 2 dÔo oikogèneie uposunìlwntou X. Lème ìti h oikogèneia 2 e�nai mia eklèptunsh (re�nement) th oikogèneia 1(gr�foume 2 � 1) e�n gia k�je B 2 2 up�rqei A 2 1 me B � A.

Basikè ènnoie : kajoliko� q¸roi kai jewr�a diast�sewn 271.2.16 Orismì . 'Estw X topologikì q¸ro kai k�lumma tou X. O megalÔtero akèraio n ètsi ¸ste to k�lumma na perièqei n+1 sÔnola me mh ken  tom  kale�tai t�xh(order) tou kalÔmmato kai sumbol�zetai me ord( ). E�n den up�rqei tètoio akèraio or�zoume ord( ) =1.Apì ton parap�nw orismì prokÔptei ìti e�n ord � n, tìte gia k�je n+2 diakekrimènastoiqe�a Ai1 ; Ai2; : : : ; Ain+2 tou kalÔmmato èqoumeAi1 \ Ai2 \ : : : \ Ain+2 = ;:Eidikìtera, e�n ord( ) = �1, tìte = f;g.H Di�stash th KalÔyew 1.2.17 Orismì . JewroÔme th sun�rthsh dim me ped�o orismoÔ thn kl�sh ìlwn twnq¸rwn kai ped�o tim¸n to sÔnolo ! [ f�1;1g pou or�zetai w ex  :dim(X) � n, ìpou n 2 ! [ f�1g, e�n gia k�je peperasmèno anoiktì k�lumma tou Xup�rqei (peperasmènh) anoikt  eklèptunsh r tou me ord(r) � n.H sun�rthsh dim kale�tai di�stash th kalÔyew ( overing dimension)   di�stashtwn �Ce h-Lebesgue.1.2.18 Je¸rhma. Gia k�je kleistì upìqwro M enì topologikoÔ q¸rou X,dim(M) � dim(X):1.2.19 Je¸rhma. (1) Gia k�je T1-q¸ro X, ind(X) � Ind(X).(2) Gia k�je fusikì q¸ro X, dim(X) � Ind(X).1.2.20 Je¸rhma. (Kat�etov-Morita) Gia k�je metrikopoi simo q¸ro X,Ind(X) = dim(X):1.2.21 Je¸rhma. Gia k�je diaqwr�simo metrikopoi simo q¸ro X,ind(X) = Ind(X) = dim(X):1.2.22 Je¸rhma. 'Estw X fusikì q¸ro . E�n up�rqei akolouj�a F1; F2; : : : ; kleist¸nuposunìlwn tou X ètsi ¸ste dim(Fi) � n gia i = 1; 2; : : : ; kai X = [1i=1Fi, tìtedim(X) � n:

28 Kef�laio 11.2.23 Je¸rhma. 'Estw X fusikì q¸ro . E�n X = X1 [X2, tìtedim(X) � dim(X1) + dim(X2) + 1:1.2.24 Je¸rhma. (The Embedding Theorem) K�je diaqwr�simo metrikopoi simo q¸-ro X me 0 � ind(X) � n emfuteÔetai ston Eukle�deio q¸ro R2n+1 .1.2.25 Je¸rhma. Gia k�je akèraio n � 0 kai plhj�rijmo � � ! h kl�sh ìlwn twnfusik¸n (Ty hono�, metrikopoi simwn) q¸rwn X me dim(X) � n kai w(X) � � èqeikajolikì q¸ro.

Kef�laio 2Kataskeu  Periektik¸n Q¸rwnSto kef�laio autì d�netai mia mèjodo kataskeu  periektik¸n q¸rwn gia mia auja�rethoikogèneia T0-q¸rwn, ìpw aut  parousi�zetai sto bibl�o [37℄ (blèpe, ep�sh , [34℄) tou kouStaÔrou Hli�dh. H mèjodo aut  e�nai sunolojewrhtik  kai qrhsimopoie�tai sta kef�laia4 kai 7 gia thn kataskeu  Kajolik¸n Q¸rwn.2.1 Prokatarktik�Se ì,ti akolouje� me � ja sumbol�zoume èna stajerì �peiro plhj�rijmo. To sÔnoloìlwn twn peperasmènwn uposunìlwn tou � sumbol�zetai me F . Eidikìtera to kenì sÔnolo; e�nai stoiqe�o tou F . Ep�sh , me th lèxh q¸ro ja ennooÔme ènan T0-q¸ro me b�ro � � .Parak�tw oi ènnoie sÔnolo, oikogèneia kai sullog  taut�zontai. Ep�sh , oi kl�sei mpore� na mhn e�nai sÔnola. 'Ena sÔnolo e�nai mia kl�sh pou e�nai stoiqe�o mia �llh kl�sh .Ja qrhsimopoioÔme to sÔmbolo <<� >> gia na eis�goume kainoÔrgiou sumbolismoÔ qwr� na k�noume lìgo gi> autì.2.1.1 Orismì . 'Ena �-diktuwmèno sÔnolo (�-indexed set)   apl� diktuwmèno sÔ-nolo e�nai mia sun�rthsh F apì èna sÔnolo � se èna sÔnolo Y . Dhlad ,F = f(�; y�) : � 2 �g;ìpou y� = F (�) gia k�je � 2 �. Epomènw , dÔo stoiqe�a (�1; y�1) kai (�2; y�2) tou F e�naidiaforetik� e�n kai mìnon e�n ta stoiqe�a �1 kai �2 tou � e�nai diaforetik�. Wstìso, pro q�rin aplìthta , k�je stoiqe�o (�; y�) tou F taut�zetai me to stoiqe�o y� tou Y kai todiktuwmèno sÔnolo F sumbol�zetai ep�sh me fy� : � 2 �g. Shmei¸noume ìti e�n y�1 = y�2gia k�poia �1; �2 2 � me �1 6= �2, tìte ta y�1 kai y�2 ja ta jewroÔme diaforetik� stoiqe�atou Y . 29

30 Kef�laio 2'Estw F : �! Y èna diktuwmèno sÔnolo. E�n F (�) = Y , tìte to diktuwmèno sÔnoloF kale�tai diktÔwsh (indi ation) tou Y .2.1.2 Orismì . 'Estw X kl�sh. Mia sqèsh isodunam�a ep� th X e�nai mia upokl�sh� tou X �X, tètoia ¸ste gia k�je x; y; z 2 X na isqÔoun:(1) (x; x) 2� (autopaj  idiìthta).(2) E�n (x; y) 2�, tìte (y; x) 2� (summetrik  idiìthta).(3) E�n (x; y) 2� kai (y; z) 2�, tìte (x; z) 2� (metabatik  idiìthta).E�n (x; y) 2� ja gr�foume ep�sh x � y. Gia k�je x 2 X h kl�sh fy 2 X : x � ygkale�tai kl�sh isodunam�a th �. E�n h kl�sh X e�nai sÔnolo, tìte to sÔnolo ìlwntwn kl�sewn isodunam�a th � sumbol�zetai me C(�).2.1.3 Orismì . 'Estw X sÔnolo. Mia mh ken  sullog  R uposunìlwn tou X kale�taidaktÔlio (ring) tou X, ìtan:(1) E�nai kleist  w pro ti peperasmène tomè . Dhlad , gia k�je jetikì akèraio n kaigia k�je U1; : : : ; Un 2 R h tom  U1 \ : : : \ Un 2 R.(2) E�nai kleist  w pro ti peperasmène en¸sei . Dhlad , gia k�je jetikì akèraio nkai gia k�je U1; : : : ; Un 2 R h ènwsh U1 [ : : : [ Un 2 R.2.1.4 Orismì . 'Estw X sÔnolo. Mia mh ken  sullog  A uposunìlwn tou X kale�tai�lgebra (algebra) tou X, ìtan:(1) E�nai kleist  w pro ta sumplhr¸mata. Dhlad , gia k�je U 2 A to sumpl rwmaU 2 A.(2) E�nai kleist  w pro ti peperasmène en¸sei . Dhlad , gia k�je jetikì akèraio nkai gia k�je U1; : : : ; Un 2 A h ènwsh U1 [ : : : [ Un 2 A.2.1.5 Parat rhsh. K�je �lgebra A enì sunìlou X e�nai kleist  w pro ti pepera-smène tomè kai perièqei ta sÔnola ; kai X.Pr�gmati, èstw U1; : : : ; Un 2 A. Tìte, U1 ; : : : ; Un 2 A. Sunep¸ ,U1 [ : : : [ Un = (U1 \ : : : \ Un) 2 A:Opìte, U1 \ : : : \ Un = ((U1 \ : : : \ Un) ) 2 A:'Estw U 2 A. Tìte, U 2 A. Sunep¸ , X = U [ U 2 A kai ; = X 2 A.

Kataskeu  Periektik¸n Q¸rwn 31Apì thn parap�nw parat rhsh e�nai fanerì ìti k�je �lgebra enì sunìlou e�nai kaidaktÔlio tou �diou sunìlou.2.1.6 Parade�gmata.(1) Gia k�je sÔnolo X h oikogèneia f;; fXgg e�nai �lgebra.(2) Gia k�je sÔnolo X to dunamosÔnolo P(X) tou X e�nai �lgebra.2.1.7 Prìtash. H tom  opoioud pote pl jou algebr¸n (ant�stoiqa, daktul�wn) enì sunìlou X e�nai ep�sh �lgebra (ant�stoiqa, daktÔlio ) tou X.2.1.8 Orismì . 'Estw X sÔnolo kai G � P(X). H tom  ìlwn twn algebr¸n (ant�stoiqa,daktul�wn) tou X pou perièqoun to G kale�tai �lgebra paragìmenh apì to G (an-t�stoiqa, daktÔlio paragìmeno apì to G). O daktÔlio pou par�getai apì to Gsumbol�zetai me G}. Profan¸ o G} e�nai o el�qisto daktÔlio tou X (w pro th sqèshtou perièqesjai) pou perièqei to G.2.1.9 Parat rhsh. 'Estw X sÔnolo kai G � P(X). JewroÔme to sÔnolo G1 = G [ G ,ìpou G = fU : U 2 Gg.(1) O el�qisto daktÔlio tou X pou perièqei to G apotele�tai apì ìle ti peperasmène en¸sei [fAi : i 2 n 2 !g, ìpou Ai = Gij1 \ : : : \ Gijki kai Gijm 2 G gia k�je i 2 n kaim = 1; : : : ; ki.(2) H el�qisth �lgebra tou X pou perièqei to G apotele�tai apì ìle ti peperasmène en¸sei [fAi : i 2 n 2 !g, ìpou Ai = Gij1 \ : : : \ Gijki kai Gijm 2 G1 gia k�je i 2 n kaim = 1; : : : ; ki.2.1.10 Orismì . 'Estw X kai Y dÔo sÔnola. Mia sun�rthsh i apì èna daktÔlio Rtou X sto dunamosÔnolo P(Y ) kale�tai omomorfismì (homomorphism), e�n gia k�jeU; V 2 R èqoume ìti:(1) i(U \ V ) = i(U) \ i(V ).(2) i(U [ V ) = i(U) [ i(V ).2.1.11 Orismì . 'Estw X kai Y dÔo sÔnola. Mia sun�rthsh i apì mia �lgebra Atou X sto dunamosÔnolo P(Y ) kale�tai omomorfismì (homomorphism), e�n gia k�jeU; V 2 A èqoume ìti:(1) i(X n U) = Y n i(U).(2) i(U [ V ) = i(U) [ i(V ).

32 Kef�laio 22.1.12 Parat rhsh. E�n stou orismoÔ 2.1.10 kai 2.1.11 h i e�nai epiplèon 1-1, tìte hant�stoiqh sun�rthsh kale�tai isomorfismì (isomorphism).2.1.13 Prìtash. 'Estw X kai Y dÔo sÔnola, A mia �lgebra tou X kai i : A ! P(Y )èna omomorfismì . Tìte, gia k�je U; V 2 A èqoume ìti i(U \ V ) = i(U) \ i(V ).2.1.14 Prìtash. 'Estw X kai Y dÔo sÔnola, A mia �lgebra tou X kai i : A ! P(Y )èna omomorfismì . Tìte, h eikìna i(A) tou A mèsw tou i e�nai mia �lgebra tou Y .2.1.15 Prìtash. 'Estw X kai Y dÔo sÔnola, A mia �lgebra tou X kai i : A ! P(Y )èna isomorfismì . Tìte, h ant�strofh sun�rthsh i�1 : i(A)!A e�nai isomorfismì .2.1.16 Prìtash. 'Estw X, Y kai Z tr�a sÔnola, A mia �lgebra touX kai i : A! P(Y ),j : i(A)! P(Z) dÔo isomorfismo�. Tìte, h sÔnjesh j Æ i : A! P(Z) e�nai isomorfismì .A i //jÆi !!CC

CC

CC

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Ci(A)j

��P(Z)2.1.17 Prìtash. 'Estw X kai Y dÔo sÔnola, A mia �lgebra tou X kai i : A ! P(Y )èna isomorfismì . Tìte, gia k�je U; V;W 2 A oi sqèsei U = ;, U = X, U \ V = Wkai U � V e�nai isodÔname twn sqèsewn i(U) = ;, i(U) = Y , i(U) \ i(V ) = i(W ) kaii(U) � i(V ), ant�stoiqa.2.1.18 Prìtash. 'Estw X kai Y dÔo sÔnola, G � P(X) kai A h el�qisth �lgebra touX pou perièqei to G. Tìte, isqÔoun ta ex  :(1) E�n i : A ! P(Y ) e�nai èna omomorfismì , tìte h i(A) e�nai h el�qisth �lgebra touY pou perièqei to i(G).(2) E�n h1 : A ! P(Y ) kai h2 : A ! P(Y ) e�nai dÔo omomorfismo� tètoioi ¸ste h1(U) =h2(U) gia k�je U 2 G, tìte h1(U) = h2(U) gia k�je U 2 A.2.1.19 Sumbolismo�. H kl�sh ìlwn twn diataktik¸n arijm¸n sumbol�zetai me O. Sthnkl�shO me (+) sumbol�zoume to fusikì �jroisma tou Hessenberg (the natural sum of Hes-senberg) (blèpe, gia par�deigma, [44℄). Shmei¸noume ti parak�tw idiìthte tou fusikoÔajro�smato :(1) �(+)� = �(+)�,(2) e�n �1 < �2, tìte �1(+)� < �2(+)� kai

Kataskeu  Periektik¸n Q¸rwn 33(3) �(+)n = � + n gia n < !.Epiplèon jewroÔme dÔo sÔmbola, <<�1>> kai <<1>> tètoia ¸ste �1 < � <1 gia k�je � 2 Okai �1(+)� = �(+)(�1) = �, 1(+)� = �(+)1 =1 gia k�je � 2 O [ f�1;1g.Gia k�je plhj�rijmo � me �+ sumbol�zoume to mikrìtero plhj�rijmo pou e�nai megalÔ-tero apì ton �. O pr¸to �peiro plhj�rijmo sumbol�zetai me !. Gia k�je sÔnolo Xme jXj sumbol�zoume ton plhj�rijmo tou X.2.2 Shmademènoi q¸roi kai prìtupe sqèsei isodu-nam�a Sthn enìthta aut  sthn kl�sh ìlwn twn shmademènwn q¸rwn, qrhsimopoi¸nta ti �lgebre twn sunìlwn, ja orisjoÔn k�poie sqèsei isodunam�a . Oi sqèsei isodunam�a autè ja pa�xoun shmantikì rìlo sthn kataskeu  twn Periektik¸n Q¸rwn.2.2.1 Orismì . 'Estw X q¸ro . K�je � -diktuwmènh b�sh fUXÆ : Æ 2 �g tou X kale�taishm�di (mark) tou q¸rou X. O q¸ro X kale�tai shmademèno (marked), e�n èqeidoje� èna shm�di tou X.2.2.2 Orismì . 'Estw X shmademèno q¸ro , fUXÆ : Æ 2 �g to ant�stoiqo shm�di touX kai s 2 F n f;g. H el�qisth �lgebra tou X pou perièqei to sÔnolo fUXÆ : Æ 2 sgkale�tai s-�lgebra tou shmademènou q¸rou X kai sumbol�zetai me AXs . Ta stoiqe�a UXÆkai X n UXÆ th AXs sumbol�zontai ep�sh me X(Æ;0) kai X(Æ;1), ant�stoiqa.2.2.3 Parat rhsh. Apì ton parap�nw orismì e�nai fanerì ìti e�n ; 6= t � s, tìteAXt � AXs .2.2.4 Sumbolismo�. 'Estw X shmademèno q¸ro kai s 2 F nf;g. Gia k�je f 2 2s, ìpou2 = f0; 1g, me X(s;f) sumbol�zoume to sÔnolo \fX(Æ;f(Æ)) : Æ 2 sg. Dhlad ,X(s;f) = \fX(Æ;f(Æ)) : Æ 2 sg:Profan¸ X(s;f) 2 AXs , w peperasmènh tom  stoiqe�wn th �lgebra AXs . Ep�sh , me 2sXsumbol�zoume to uposÔnolo tou 2s pou apotele�tai apì ìla ta stoiqe�a f tou 2s gia taopo�a X(s;f) 6= ;. Dhlad , 2sX = ff 2 2s : X(s;f) 6= ;g:Tèlo , gia k�je Æ 2 s jètoumeu(X; s; Æ) = ff 2 2sX : f(Æ) = 0g:

34 Kef�laio 22.2.5 Prìtash. 'Estw X shmademèno q¸ro kai s 2 F n f;g. Tìte, isqÔoun ta ex  :(1) Gia k�je Æ 2 s, X(Æ;0) \X(Æ;1) = ; kai X(Æ;0) [X(Æ;1) = X.(2) Gia k�je f; g 2 2sX me f 6= g, isqÔei X(s;f) \X(s;g) = ;.(3) [fX(s;f) : f 2 2sXg = X.2.2.6 Sumbolismo�. 'Estw X shmademèno q¸ro kai s 2 F n f;g. Apì thn parap�nwprìtash prokÔptei ìti ta sÔnola X(s;f), f 2 2sX , apoteloÔn mia diamèrish tou sunìlou X.Epomènw , gia k�je x 2 X up�rqei monadik  sun�rthsh f 2 2sX tètoia ¸ste x 2 X(s;f).Sumbol�zoume me dXs th sun�rthsh apì to X sto 2sX pou or�zetai w ex  :dXs (x) = f:E�n X = ;, tìte upojètoume ìti dXs (X) = ;. Profan¸ , 2sX = dXs (X).Gia k�je u � 2sX me X(s;u) sumbol�zoume to sÔnolo [fX(s;f) : f 2 ug. Dhlad ,X(s;u) = [fX(s;f) : f 2 ug:Profan¸ X(s;u) 2 AXs , w peperasmènh ènwsh stoiqe�wn th �lgebra AXs . Sumbol�zoumeme isX th sun�rthsh apì to P(2sX) sto AXs pou or�zetai w ex  :isX(u) = X(s;u):2.2.7 Parat rhsh. 'Estw s 2 F n f;g, Æ 2 s kai u; v 2 P(2sX). Tìte, isqÔoun ta ex  :(1) X nX(s;u) = X(s;2sXnu).(2) X(s;u) [X(s;v) = X(s;u[v).(3) X(s;u(X;s;Æ)) = X(Æ;0).2.2.8 Prìtash. 'Estw X shmademèno q¸ro kai s 2 F nf;g. H sun�rthsh isX e�nai èna isomorfismì apì thn �lgebra P(2sX) ep� th �lgebra AXs tètoio ¸ste gia k�je Æ 2 s,isX(u(X; s; Æ)) = X(Æ;0).2.2.9 Parat rhsh. H P(2sX) e�nai h el�qisth �lgebra ep� tou 2sX pou perièqei ta sÔnolau(X; s; Æ) = ff 2 2sX : f(Æ) = 0g. Pr�gmati, epeid  ta stoiqe�a tou 2sX pa�rnoun mìno dÔotimè , e�te mhdèn e�te èna, k�je �lgebra A pou perièqei ta sÔnola u(X; s; Æ) ja isoÔtaianagkastik� me to dunamosÔnolo P(2sX). Gia par�deigma, èstw s = fÆ1; Æ2; Æ3; Æ4g kaig 2 2sX me timè g(Æ1) = 0, g(Æ2) = 0, g(Æ3) = 1, g(Æ4) = 0. Tìte,fgg = u(X; s; Æ1) \ u(X; s; Æ2) \ (2sX n u(X; s; Æ3)) \ u(X; s; Æ4) 2 A:

Kataskeu  Periektik¸n Q¸rwn 35Opìte, e�n g; h 2 2sX , tìte fgg; fhg 2 A kai sunep¸ fg; hg = fgg [ fhg 2 A.2.2.10 Parat rhsh. H isX e�nai monadik . Dhlad , e�n h e�nai èna isomorfismì apì thn�lgebra P(2sX) ep� th �lgebra AXs tètoio ¸ste gia k�je Æ 2 s, h(u(X; s; Æ)) = X(Æ;0),tìte h = isX . Pr�gmati, epeid  h P(2sX) e�nai h el�qisth �lgebra ep� tou 2sX pou perièqei tasÔnola u(X; s; Æ) kai gia k�je u(X; s; Æ) 2 P(2sX) èqoume ìti h(u(X; s; Æ)) = isX(u(X; s; Æ)),apì thn Prìtash 2.1.18(2) prokÔptei ìti h(v) = isX(v) gia k�je v 2 P(2sX).2.2.11 Orismì . Gia k�je s 2 F n f;g sthn kl�sh ìlwn twn shmademènwn q¸rwnor�zoume mia sqèsh isodunam�a �sm w ex  : DÔo shmademènoi q¸roi X kai Y e�nai �sm-isodÔnamoi (gr�foume X �sm Y ) tìte kai mìno tìte ìtan up�rqei èna isomorfismì iapì thn �lgebra AXs ep� th �lgebra AYs tètoio ¸ste i(X(Æ;0)) = Y(Æ;0) gia k�je Æ 2 s. Hsqèsh isodunam�a �sm kale�tai s-prìtuph (s-standard). O isomorfismì i, pou apì thnPrìtash 2.1.18(2) e�nai monos manta kajorismèno , kale�tai fusikì (natural).2.2.12 Parat rhsh. E�n ; 6= t � s, tìte �sm��tm. Pr�gmati, èstw (X; Y ) 2�sm.ApodeiknÔoume ìti (X; Y ) 2�tm. Epeid  (X; Y ) 2�sm, up�rqei èna isomorfismì i apìthn �lgebra AXs ep� th �lgebra AYs tètoio ¸ste i(X(Æ;0)) = Y(Æ;0) gia k�je Æ 2 s.JewroÔme ton periorismì ijAXt : AXt ! AYt tou i ep� th �lgebra AXt . O ijAXt e�naiisomorfismì kai epiplèon gia k�je Æ 2 t � s isqÔei ijAXt (X(Æ;0)) = Y(Æ;0). Dhlad , o ijAXte�nai o fusikì isomorfismì apì thn �lgebra AXt sthn �lgebra AYt . 'Ara, (X; Y ) 2�tm.2.2.13 Prìtash. 'Estw X kai Y dÔo shmademènoi q¸roi kai s 2 F n f;g. Oi parak�twsunj ke e�nai isodÔname .(1) X �sm Y .(2) 2sX = 2sY .(3) u(X; s; Æ) = u(Y; s; Æ) gia k�je Æ 2 s.2.3 Oi Periektiko� Q¸roi T(M;R)Se ì,ti akolouje� sthn enìthta aut  upojètoume ìti d�netai mia auja�reth mh ken diktuwmènh oikogèneia S shmademènwn topologik¸n q¸rwn.2.3.1 Orismì . 'Estw R0 � f�s0: s 2 Fg kai R1 � f�s1: s 2 Fg dÔo F -diktuwmène oikogèneie apì sqèsei isodunam�a ep� th S. Lème ìti h oikogèneia R1 e�nai telik¸ leptìterh (�nal re�nement) th oikogèneia R0, e�n gia k�je s 2 F up�rqei t 2 F tètoio¸ste �t1��s0.

36 Kef�laio 22.3.2 Parat rhsh. 'Estw R0 � f�s0: s 2 Fg, R1 � f�s1: s 2 Fg kai R2 � f�s2: s 2 Fgtre� F -diktuwmène oikogèneie apì sqèsei isodunam�a ep� th S. E�n h oikogèneia R0e�nai telik¸ leptìterh th R1 kai h oikogèneia R1 e�nai telik¸ leptìterh th R2, tìte hoikogèneia R0 e�nai telik¸ leptìterh th R2 (metabatik  idiìthta).Pr�gmati, èstw s 2 F . Epeid  h R1 e�nai telik¸ leptìterh th R2, up�rqei p 2 Ftètoio ¸ste �p1��s2. Ep�sh , epeid  h R0 e�nai telik¸ leptìterh th R1, up�rqei t 2 Ftètoio ¸ste �t0��p1. Sunep¸ , �t0��p1��s2 kai epomènw �t0��s2.2.3.3 Orismì . Mia F -diktuwmènh oikogèneia R � f�s: s 2 Fg apì sqèsei isodunam�a ep� th S kale�tai epitrept  (admissible), e�n ikanopoioÔntai oi parak�tw sunj ke :(1) E�n s; t 2 F me s � t, tìte �t��s.(2) Gia k�je s 2 F o arijmì twn kl�sewn isodunam�a th sqèsh �s e�nai peperasmèno .(3) E�n s = ;, tìte �s= S� S.2.3.4 Orismì . Gia k�je X 2 S me M(X) sumbol�zoume èna shm�di tou q¸rou X. TosÔnolo M � fM(X) : X 2 Sgìlwn aut¸n twn shmadi¸n kale�tai sun-shm�di ( o-mark) th oikogèneia S.H sqèsh �sm th prohgoÔmenh enìthta or�sthke sthn kl�sh ìlwn twn shmademènwnT0-q¸rwn me b�ro � � .2.3.5 Orismì . 'Estw M � fM(X) : X 2 Sg èna sun-shm�di th oikogèneia S. Giak�je s 2 F n f;g sthn oikogèneia S or�zoume mia sqèsh isodunam�a �sm w ex  : DÔoshmademènoi q¸roi X kai Y th S e�nai �sM-isodÔnamoi (gr�foume X �sM Y ) tìte kaimìno tìte ìtan X �sm Y . Me �lla lìgia �sM=�sm \(S�S). Ep�sh , or�zoume �;M= S�S.Dhlad , e�n s = ;, tìte �sM= S� S. H F -diktuwmènh oikogèneiaRM � f�sM: s 2 Fgkale�tai M-prìtuph (M-standard).2.3.6 Parat rhsh. H M-prìtuph oikogèneia RM e�nai epitrept . Pr�gmati, ikano-poioÔntai oi tre� idiìthte tou OrismoÔ 2.3.3:(1) 'Estw s; t 2 F n f;g me s � t. Tìte, �tm��sm kai sunep¸ �tM��sM. Ep�sh , e�ns = ;, tìte profan¸ �tM��sM= S� S.

Kataskeu  Periektik¸n Q¸rwn 37(2) E�n s = ;, tìte �sM= S� S kai �ra o arijmì twn kl�sewn isodunam�a th �sM e�naipeperasmèno (h mình kl�sh isodunam�a e�nai h S). E�n s 2 F n f;g, tìte, sÔmfwna methn Prìtash 2.2.13, k�je sÔnolo 2sX , X 2 S, or�zei mia kl�sh isodunam�a th sqèsh �sM. Sunep¸ , epeid  ta sÔnola 2sX , X 2 S, e�nai uposÔnola tou peperasmènou sunìlou2s, èqoume to zhtoÔmeno.(3) Ex orismoÔ.2.3.7 Orismì . Mia F -diktuwmènh oikogèneia R � f�s: s 2 Fg apì sqèsei isodunam�a ep� th S kale�tai M-epitrept  (M-admissible), e�n e�nai epitrept  kai epiplèon e�naitelik¸ leptìterh th oikogèneia RM. Dhlad , ìtan gia k�je s 2 F up�rqei t 2 Ftètoio ¸ste �t��sM.2.3.8 Parat rhsh. Shmei¸noume ìti e�n �t��sM kai X �t Y , tìte X �sM Y kai�ra up�rqei èna isomorfismì i apì thn �lgebra AXs ep� th �lgebra AYs tètoio ¸stei(UXÆ ) = UYÆ gia k�je Æ 2 s.2.3.9 Sumbolismì . 'Estw R � f�s: s 2 Fg mia F -diktuwmènh oikogèneia apì sqèsei isodunam�a ep� th S. Me C(R) sumbol�zoume to sÔnolo [fC(�s) : s 2 Fg. Dhlad ,C(R) = [fC(�s) : s 2 Fg:Ep�sh , to daktÔlio C(R)} ja to sumbol�zoume me C}(R). Profan¸ , C}(R) � P(S).2.3.10 Parat rhsh. E�n h oikogèneia R e�nai epitrept , tìte o daktÔlio C}(R) e�naikleistì kai w pro ta sumplhr¸mata. Dhlad , e�n H 2 C}(R), tìte S nH 2 C}(R).2.3.11 Sumfwn�a. Se ì,ti akolouje� sthn enìthta aut  upojètoume ìti d�netai ènaauja�reto sun-shm�di M � fM(X) � fUXÆ : Æ 2 �g : X 2 Sgth oikogèneia S. Ep�sh , upojètoume ìti d�netai mia M-epitrept  F -diktuwmènh oikogè-neia R � f�s: s 2 Fgapì sqèsei isodunam�a ep� th S.Gia to sun-shm�di M th S kai gia thn oikogèneia R ja kataskeu�soume ènan Pe-riektikì Q¸ro T(M;R). O q¸ro autì or�zetai monos manta apì thn oikogèneia S, tosun-shm�di M th S kai thn oikogèneia R. Gia k�je stoiqe�o X th S up�rqei mia fusik topologik  emfÔteush tou X sto q¸ro T(M;R).

38 Kef�laio 22.3.12 Orismì . Sto sÔnolo ìlwn twn zeug¸n (x;X), ìpou X 2 S kai x 2 X, or�zoumemia sqèsh isodunam�a �RM w ex  : DÔo zeÔgh (x;X) kai (y; Y ) e�nai �RM-isodÔnama(gr�foume (x;X) �RM (y; Y )) tìte kai mìno tìte ìtan gia k�je s 2 F nf;g èqoume X �s Ykai dXs (x) = dYs (y).2.3.13 Parat rhsh. H sunj kh dXs (x) = dYs (y) gia k�je s 2 F nf;g e�nai isodÔnamh meth sunj kh: Gia k�je Æ 2 � e�te x 2 UXÆ kai y 2 UYÆ e�te x =2 UXÆ kai y =2 UYÆ .2.3.14 Sumbolismo�. To sÔnolo ìlwn twn kl�sewn isodunam�a th �RM sumbol�zetaime T(M;R)   apl¸ me T (an den up�rqei k�nduno sÔgqush ). Dhlad ,T = T(M;R) = C(�RM):Ep�sh , or�zoume T = ; e�n ìla ta stoiqe�a th S e�nai ken�.Gia k�je H 2 C}(R) (eidikìtera, gia k�je H 2 C(R)) to sÔnolo ìlwn twn stoiqe�wna tou T gia ta opo�a up�rqei èna stoiqe�o (x;X) tou a tètoio ¸ste X 2 H sumbol�zetaime T(M;R;H)   apl¸ me T(H) (an den up�rqei k�nduno sÔgqush ). Dhlad ,T(H) = fa 2 T : up�rqei (x;X) 2 a me X 2 Hg:2.3.15 Parat rhsh. 'Estw t 2 F kai H 2 C(�t). Tìte, to sÔnolo T(H) sump�ptei meto sÔnolo ìlwn twn a 2 T ètsi ¸ste gia k�je (x;X) 2 a, X 2 H. Dhlad ,T(H) = fa 2 T : gia k�je (x;X) 2 a; X 2 Hg:Pr�gmati, èstw a 2 T(H) kai (x;X) 2 a. Epeid  a 2 T(H), up�rqei (y; Y ) 2 a tètoio¸ste Y 2 H. E�n t 2 F n f;g, tìte epeid  (x;X); (y; Y ) 2 a, èqoume ìti X �t Y . E�nt = ;, tìte �t= S� S kai epomènw X �t Y . Sunep¸ , epeid  Y 2 H 2 C(�t), X 2 H.2.3.16 Sumbolismo�. 'Estw s; t 2 F , s 6= ;, �t��sM kai H 2 C(�t). Upojètoume ìtièna stoiqe�o X touH èqei epilege�. Tìte, ta sÔnola u(X; s; Æ) kai 2sX e�nai anex�rthta apìthn epilog  tou sunìlou X. Pr�gmati, èstw X; Y 2 H. Tìte, (X; Y ) 2�t kai sunep¸ (X; Y ) 2�sM. 'Ara, apì thn Prìtash 2.2.13, èqoume ìti 2sX = 2sY kai u(X; s; Æ) = u(Y; s; Æ).Gia k�je X 2 H me 2sH sumbol�zoume to sÔnolo 2sX kai me u(H; s; Æ) to sÔnolou(X; s; Æ).Gia k�je u 2 P(2sH) me T(s;u)(H) sumbol�zoume to sÔnolo ìlwn twn stoiqe�wn a tou Tgia ta opo�a up�rqei èna stoiqe�o (x;X) tou a me X 2 H kai x 2 X(s;u). Dhlad ,T(s;u)(H) = fa 2 T : up�rqei (x;X) 2 a me X 2 H kai x 2 X(s;u)g:

Kataskeu  Periektik¸n Q¸rwn 39E�n u = u(H; s; Æ) gia k�poio Æ 2 s (kai sunep¸ , apì thn Prìtash 2.2.8, X(s;u) = X(Æ;0)gia k�je X 2 H), tìte to sÔnolo T(s;u)(H) sumbol�zetai ep�sh me T(Æ;0)(H). Dhlad ,T(Æ;0)(H) = fa 2 T : up�rqei (x;X) 2 a me X 2 H kai x 2 X(Æ;0)g:Me AHs sumbol�zoume to sÔnolo pou èqei w stoiqe�a ta sÔnola T(s;u)(H), u 2 P(2sH).Dhlad , AHs = fT(s;u)(H) : u 2 P(2sH)g:Tèlo , me isH sumbol�zoume th sun�rthsh apì to P(2sH) sto AHs pou or�zetai w ex  :isH(u) = T(s;u)(H):2.3.17 Prìtash. 'Estw s; t 2 F , s 6= ;, �t��sM, H 2 C(�t) kai u 2 P(2sH). To sÔnoloT(s;u)(H) sump�ptei me to sÔnolo ìlwn twn a 2 T ètsi ¸ste gia k�je (x;X) 2 a na èqoumeX 2 H kai x 2 X(s;u). Dhlad ,T(s;u)(H) = fa 2 T : gia k�je (x;X) 2 a èqoume X 2 H kai x 2 X(s;u)g:2.3.18 Parat rhsh. 'Estw s; t 2 F , s 6= ;, �t��sM, H 2 C(�t) kai u; v 2 P(2sH).Tìte, isqÔoun ta ex  :(1) T(H) n T(s;u)(H) = T(s;2sHnu)(H):(2) T(s;u)(H) [ T(s;v)(H) = T(s;u[v)(H):2.3.19 Prìtash. 'Estw s; t 2 F , s 6= ;, �t��sM kai H 2 C(�t).(1) To sÔnolo AHs e�nai mia �lgebra tou T(H).(2) H sun�rthsh isH e�nai èna isomorfismì apì thn �lgebra P(2sH) ep� th �lgebra AHs .2.3.20 Prìtash. 'Estw s; t 2 F , s 6= ;, �t��sM, H 2 C(�t) kai X 2 H. Tìte, hsun�rthsh i = isH Æ (isX)�1 e�nai èna isomorfismì apì thn �lgebra AXs ep� th �lgebra AHs tètoio ¸ste i(X(Æ;0)) = T(Æ;0)(H) gia k�je Æ 2 s.AXs (isX)�1//i&&NNNNNNNNNNNNN

P (2sX) = P (2sH)isH��AHs

40 Kef�laio 22.3.21 Sumbolismo�. Gia k�je Æ 2 � kai H 2 C}(R) (eidikìtera, gia k�je H 2 C(R))me UTÆ (H) sumbol�zoume to sÔnolo ìlwn twn stoiqe�wn a tou T gia ta opo�a up�rqei ènastoiqe�o (x;X) tou a me X 2 H kai x 2 UXÆ . Dhlad ,UTÆ (H) = fa 2 T : up�rqei (x;X) 2 a me X 2 H kai x 2 UXÆ g:Sunep¸ , e�n gia k�poio stoiqe�o s tou F n f;g èqoume Æ 2 s kai H 2 C(�t), ìpou t 2 Fkai �t��sM, tìte UTÆ (H) = T(Æ;0)(H) 2 AHs .Gia k�je � � � kai L 2 C}(R) jètoume:(1) BT = fUTÆ (H) : Æ 2 � kai H 2 C(R)g.(2) BL = fUTÆ (H) 2 BT : H � Lg.(3) BT� = fUTÆ (H) 2 BT : Æ 2 �g.(4) BL� = fUTÆ (H) 2 BT� : H � Lg.(5) BT} = fUTÆ (H) : Æ 2 � kai H 2 C}(R)g.(6) BL} = fUTÆ (H) 2 BT} : H � Lg.(7) BT};� = fUTÆ (H) 2 BT} : Æ 2 �g.(8) BL};� = fUTÆ (H) 2 BT};� : H � Lg.2.3.22 Parat rhsh. 'Estw H 2 C}(R). Tìte,T(H) = [fUTÆ (H) : Æ 2 �g:2.3.23 Parat rhsh. 'Estw L 2 C}(R). Tìte,BL};� = fU \ T(L) : U 2 BT};�g:Shmei¸noume ìti, genik�, BL� 6= fU \ T(L) : U 2 BT�g.2.3.24 Parat rhsh. Epeid  jFj � � kai gia k�je s 2 F o arijmì twn kl�sewnisodunam�a th sqèsh �s e�nai peperasmèno , jC(R)j � � . Autì shma�nei ìti jBTj � � .An�loga, èqoume ìti jC}(R)j � � kai sunep¸ jBTj � � .2.3.25 Prìtash. (1) To sÔnolo BT e�nai b�sh gia mia topolog�a ep� tou T. Epiplèon,e�n � e�nai èna uposÔnolo tou � ètsi ¸ste gia k�je X 2 S to sÔnolo fUXÆ : Æ 2 �g nae�nai b�sh tou X, tìte to sÔnolo BT� e�nai b�sh gia thn �dia topolog�a ep� tou T.

Kataskeu  Periektik¸n Q¸rwn 41(2) Gia k�je L 2 C}(R) to sÔnolo BL e�nai b�sh tou upoq¸rou T(L) tou T. Epiplèon,e�n � e�nai èna uposÔnolo tou � ètsi ¸ste gia k�je X 2 S to sÔnolo fUXÆ : Æ 2 �g nae�nai b�sh tou X, tìte to sÔnolo BL� e�nai b�sh tou T(L).2.3.26 Orismì . Oi b�sei BT kai BL kaloÔntai prìtupe (standard). Gia k�je � � �oi b�sei BT� kai BL� kaloÔntai �-prìtupe (�-standard).2.3.27 Orismì . 'Estw T o topologikì q¸ro pou èqei b�sh to sÔnolo BT. O Tkale�tai Periektikì Q¸ro (Containing Spa e) th oikogèneia S pou antistoiqe� stosun-shm�di M � fM(X) � fUXÆ : Æ 2 �g : X 2 Sg kai sthn oikogèneia R � f�s: s 2 Fg.Epeid  jBTj � � , èqoume ìti w(T) � � .H epìmenh prìtash ma lèei ìti ta stoiqe�a tou sunìlou BT} e�nai anoikt� sto T.Sunep¸ , epeid  BT � BT}, to BT} apotele� kai autì b�sh gia to T. Ep�sh , gnwr�zoumeìti jBT}j � � .2.3.28 Prìtash. 'Estw Æ 2 � kai H 2 C}(R). Oi parak�tw prot�sei e�nai alhje� .(1) Up�rqei t 2 F ètsi ¸ste to H na e�nai ènwsh kl�sewn isodunam�a th sqèsh �t.Epiplèon, e�n t � q 2 F , tìte h H e�nai ep�sh ènwsh kl�sewn isodunam�a th sqèsh �q.(2) To sÔnolo UTÆ (H) e�nai anoiktì sto T kaiUTÆ (H) = fa 2 T : gia k�je (x;X) 2 a èqoume ìti X 2 H kai x 2 UXÆ g:(3) To sÔnolo T(H) e�nai anoiktì kai kleistì sto T.(4) 'EqoumeClT(UTÆ (H)) = fa 2 T : up�rqei (x;X) 2 a me X 2 H kai x 2 ClX(UXÆ )g= fa 2 T : gia k�je (x;X) 2 a; X 2 H kai x 2 ClX(UXÆ )g:(5) 'EqoumeBdT(UTÆ (H)) = fa 2 T : up�rqei (x;X) 2 a me X 2 H kai x 2 BdX(UXÆ )g= fa 2 T : gia k�je (x;X) 2 a; X 2 H kai x 2 BdX(UXÆ )g:2.3.29 Prìtash. (1) To sÔnolo BT} e�nai b�sh tou q¸rou T. Epiplèon, e�n � e�nai ènauposÔnolo tou � ètsi ¸ste gia k�je X 2 S to sÔnolo fUXÆ : Æ 2 �g na e�nai b�sh tou X,tìte to sÔnolo BT};� e�nai ep�sh b�sh tou T.

42 Kef�laio 2(2) Gia k�je L 2 C}(R) to sÔnolo BL} e�nai b�sh tou upoq¸rou T(L) tou T. Epiplèon,e�n � e�nai èna uposÔnolo tou � ètsi ¸ste gia k�je X 2 S to sÔnolo fUXÆ : Æ 2 �g nae�nai b�sh tou X, tìte to sÔnolo BL};� e�nai b�sh tou T(L).2.3.30 Orismì . Oi b�sei BT} kai BL} kaloÔntai }-prìtupe (}-standard). Gia k�je� � � oi b�sei BL};� kaloÔntai (}; �)-prìtupe ((}; �)-standard).2.3.31 Prìtash. O q¸ro T e�nai T0-q¸ro me b�ro w(T) � � .2.3.32 Sumbolismì . 'Estw X 2 S. Tìte gia k�je x 2 X up�rqei monadikì shme�oa 2 T tètoio ¸ste (x;X) 2 a. Sumbol�zoume me eXT thn apeikìnish apì to q¸ro X stoq¸ro T pou or�zetai w ex  : eXT (x) = a:2.3.33 Prìtash. Gia k�je X 2 S h apeikìnish eXT e�nai mia emfÔteush tou X sto T.2.3.34 Orismì . H apeikìnish eXT kale�tai fusik  emfÔteush (natural embedding) touX sto T.2.3.35 Parat rhsh. 'Eqoume T = [feXT (X) : X 2 Sg.2.4 Idi�zonta uposÔnola twn Periektik¸n Q¸rwnSe ì,ti akolouje� sthn enìthta aut  upojètoume ìti èqoun doje�:(1) Mia auja�reth mh ken  diktuwmènh oikogèneia S shmademènwn topologik¸n q¸rwn.(2) 'Ena auja�reto sun-shm�diM � fM(X) � fUXÆ : Æ 2 �g : X 2 Sgth oikogèneia S.Ep�sh , upojètoume ìti gia k�je X 2 S èna uposÔnolo QX tou X e�nai dosmèno.2.4.1 Orismì . To sÔnolo fQX : X 2 Sg kale�tai periorismì (restri tion) th oiko-gèneia S kai sumbol�zetai me Q. Dhlad ,Q = fQX : X 2 Sg:To sÔnolo QX sumbol�zetai ep�sh me Q(X). Profan¸ , to Q e�nai mia S-diktuwmènhoikogèneia apì q¸rou .

Kataskeu  Periektik¸n Q¸rwn 432.4.2 Orismì . 'Estw X 2 S. H � -diktuwmènh b�sh fUXÆ \QX : Æ 2 �g tou QX kale�tai�qno (tra e) tou M(X) ep� tou QX kai sumbol�zetai me M(X)QX . Dhlad ,M(X)QX = fUQXÆ � UXÆ \QX : Æ 2 �g:2.4.3 Prìtash. 'Estw X 2 S. Gia k�je s 2 F n f;g kai x 2 QX èqoumedXs (x) = dQXs (x):Sunep¸ , dXs (QX) = dQXs (QX).2.4.4 Orismì . To sun-shm�di fM(X)QX : QX 2 Qg tou Q kale�tai �qno (tra e) touM ep� tou Q kai sumbol�zetai me MjQ. Dhlad ,MjQ = fM(X)QX : QX 2 Qg= ffUQXÆ : Æ 2 �g : QX 2 Qg= ffUXÆ \QX : Æ 2 �g : QX 2 Qg= ffUXÆ \QX : Æ 2 �g : X 2 Sg:2.4.5 Orismì . 'Estw� mia sqèsh isodunam�a ep� tou S. Or�zoume mia sqèsh isodunam�a �jQ ep� tou Q w ex  : DÔo stoiqe�a QX kai QY tou Q e�nai �jQ-isodÔnama (gr�foumeQX �jQ QY ), tìte kai mìno tìte, ìtan X � Y . H sqèsh isodunam�a �jQ kale�tai �qno (tra e) th � ep� tou Q.2.4.6 Parat rhsh. 'Estw �1 kai �2 dÔo sqèsei isodunam�a ep� th S. E�n �1��2,tìte �1jQ � �2jQ.2.4.7 Orismì . 'Estw R � f�s: s 2 Fg mia F -diktuwmènh oikogèneia apì sqèsei isodunam�a ep� th S. H F -diktuwmènh oikogèneia f�sjQ : s 2 Fg apì sqèsei isodunam�a ep� tou Q kale�tai �qno (tra e) th R ep� tou Q kai sumbol�zetai me RjQ. Dhlad ,RjQ = f�sjQ : s 2 Fg:2.4.8 Parat rhsh. 'Estw R � f�s: s 2 Fg mia F -diktuwmènh oikogèneia apì sqèsei isodunam�a ep� th S. E�n h R e�nai epitrept , tìte kai h RjQ e�nai epitrept .2.4.9 Parat rhsh. 'Estw R0 kai R1 dÔo F -diktuwmène oikogèneie apì sqèsei isodu-nam�a ep� th S. E�n h R0 e�nai telik¸ leptìterh th R1, tìte kai h R0jQ e�nai telik¸ leptìterh th R1jQ.

44 Kef�laio 22.4.10 Orismì . 'Estw H 2 C}(R). To sÔnolo fQX 2 Q : X 2 Hg kale�tai �qno (tra e) tou H ep� tou Q kai sumbol�zetai me HjQ. Dhlad ,HjQ = fQX 2 Q : X 2 Hg:2.4.11 Parat rhsh.(1) E�n H 2 C}(R), tìte HjQ 2 C}(RjQ).(2) E�n H 2 C(R), tìte HjQ 2 C(RjQ).2.4.12 Parat rhsh. Shmei¸noume ìti, en gènei, to �qno th M-prìtuph oikogèneia RM � f�sM: s 2 Fgep� tou Q den e�nai �so me thn MjQ-prìtuph oikogèneiaRMjQ � f�sMjQ: s 2 Fg:Dhlad , RMjQ 6= RMjQ:To gegonì autì dikaiologe� ton parak�tw orismì.2.4.13 Orismì . Mia F -diktuwmènh oikogèneia R � f�s: s 2 Fg apì sqèsei isodu-nam�a ep� th S kale�tai (M;Q)-epitrept  ((M;Q)-admissible), e�n e�nai M-epitrept kai epiplèon to �qno RjQ th R ep� tou Q e�nai MjQ-epitrept  oikogèneia apì sqèsei isodunam�a ep� tou Q.2.4.14 Prìtash. 'Estw R � f�s: s 2 Fg mia M-epitrept  oikogèneia apì sqèsei isodunam�a ep� th S. H R e�nai (M;Q)-epitrept  e�n kai mìnon e�n gia k�je s 2 F nf;gup�rqei t 2 F n f;g ètsi ¸ste gia k�je X; Y 2 S na isqÔei h sunepagwg X �t Y ) dXs (QX) = dYs (QY ):2.4.15 Par�deigma. H oikogèneia R0 � f�s0: s 2 Fg, ìpou X �s0 Y e�n kai mìnon e�nX �sM Y kai dXs (QX) = dYs (QY ), e�nai (M;Q)-epitrept .2.4.16 Sumbolismo�. 'Estw R mia (M;Q)-epitrept  oikogèneia apì sqèsei isodunam�a ep� th S. Tìte, ektì apì ton Periektikì Q¸ro T(M;R) mporoÔme ep�sh na jewr soume

Kataskeu  Periektik¸n Q¸rwn 45ton Periektikì Q¸ro T(MjQ;RjQ) th diktuwmènh oikogèneia Q pou antistoiqe� stosun-shm�di MjQ � ffUQXÆ � UXÆ \QX : Æ 2 �g : QX 2 Qgkai sthn MjQ-epitrept  oikogèneiaRjQ � f�sjQ : s 2 Fg:O Periektikì Q¸ro T(MjQ;RjQ) sumbol�zetai ep�sh me TjQ.'Estw H 2 C}(R) kai HjQ to �qno tou H ep� tou Q. Tìte, mporoÔme na jewr sou-me ton upìqwro T(MjQ;RjQ;HjQ) tou T(MjQ;RjQ) ton opo�o sumbol�zoume ep�sh meT(HjQ). Ep�sh , gia k�je Æ 2 � mporoÔme na jewr soume to sÔnoloUTjQÆ (HjQ) � fa 2 TjQ : up�rqei (x;QX) 2 a me QX 2 HjQ kai x 2 UQXÆ g:Gia ton Periektikì Q¸ro TjQ h prìtuph, h �-prìtuph, h }-prìtuph kai h (}; �)-prìtuph b�sh, ìpou � � � , sumbol�zontai ant�stoiqa me BTjQ, BTjQ� , BTjQ} kai BTjQ};� . E�nLjQ 2 C}(RjQ), tìte me BLjQ , BLjQ� , BLjQ} kai BLjQ};� sumbol�zoume ti ant�stoiqe b�sei tou upoq¸rou T(HjQ) tou TjQ.2.4.17 Sumfwn�a. Se ì,ti akolouje� upojètoume ìti èqei doje� mia (M;Q)-epitrept F -diktuwmènh oikogèneia R � f�s: s 2 Fgapì sqèsei isodunam�a ep� th S.2.4.18 Prìtash. Gia k�je b 2 TjQ up�rqei èna kai mìnon èna a 2 T tètoio ¸ste giak�je x 2 QX na isqÔei h isodunam�a(x;QX) 2 b, (x;X) 2 a:2.4.19 Sumbolismì . Sumbol�zoume me eTjQT thn apeikìnish apì to q¸ro TjQ sto q¸roT pou or�zetai w ex  : eTjQT (b) = a:2.4.20 Prìtash. 'Estw Æ 2 � , H 2 C}(R) kai HjQ to �qno tou H ep� tou Q. Tìte,b 2 UTjQÆ (HjQ), a � eTjQT (b) 2 UTÆ (H):2.4.21 Prìtash. H apeikìnish eTjQT e�nai mia emfÔteush tou q¸rou TjQ sto q¸ro T.

46 Kef�laio 22.4.22 Orismì . H apeikìnish eTjQT kale�tai fusik  emfÔteush (natural embedding)tou T(MjQ;RjQ) sto T(M;R).2.4.23 Parat rhsh. Apì thn Prìtash 2.4.21 prokÔptei ìti oi q¸roi TjQ kai eTjQT (TjQ)e�nai omoiìmorfoi. Sunep¸ , topologik� oi q¸roi auto� diafèroun mìno sthn onomas�atwn stoiqe�wn tou . Taut�zoume loipìn to q¸ro TjQ me thn eikìna tou mèsw th eTjQT kaijewroÔme ìti o q¸ro TjQ e�nai upìqwro tou T. Dhlad ,TjQ � eTjQT (TjQ) � T:Ep�sh , jewroÔme ìti h emfÔteush eTjQT e�nai h tautotik  emfÔteush tou TjQ sto T.2.4.24 Orismì . To uposÔnolo TjQ tou T kale�tai idi�zwn (spe i� ) uposÔnolo touPeriektikoÔ Q¸rou T(M;R).2.4.25 Parat rhsh. 'Estw X 2 S kai eQXX h tautotik  emfÔteush tou QX sto X. Tìte,eTjQT Æ eQXTjQ = eXT Æ eQXX :QX eQXTjQ //eQXX�� ""D

DD

DD

DD

D

TjQeTjQT��X eXT // T2.4.26 Prìtash. 'Estw H 2 C}(R). Ta parak�tw e�nai alhj .(1) TjQ = [feXT (QX) : X 2 Sg.(2) T(HjQ) = [feXT (QX) : X 2 Hg.2.4.27 Prìtash. 'Estw Æ 2 � kai H 2 C}(R). Ta parak�tw e�nai alhj .(1) T(MjQ;RjQ;HjQ) = T(MjQ;RjQ) \ T(H).(2) TjQ \ UTÆ (H) = UTjQÆ (HjQ).2.4.28 Orismì . 'Ena periorismì F � fFX : X 2 Sg th S kale�tai (M;R)-pl rh periorismì ((M;R)- omplete restri tion), e�n h oikogèneia R e�nai (M;F)-epitrept  kaito uposÔnolo TjF tou T ikanopoie� thn parak�tw sunj kh: gia k�je a 2 TjF kai gia k�je(x;X) 2 a èqoume x 2 FX .

Kataskeu  Periektik¸n Q¸rwn 472.4.29 Sumbolismo�. JewroÔme tou parak�tw periorismoÔ th S:Cl(Q) � fClX(QX) : X 2 Sg,Bd(Q) � fBdX(QX) : X 2 Sg,Int(Q) � fIntX(QX) : X 2 Sg kaiCo(Q) � fX nQX : X 2 Sg.2.4.30 Orismì . 'Ena periorismì F � fFX : X 2 Sg th S kale�tai kleistì (an-t�stoiqa, anoiktì ), e�n gia k�je X 2 S to FX e�nai kleistì (ant�stoiqa, anoiktì)uposÔnolo tou X.2.4.31 Prìtash. Upojètoume ìti o Q e�nai kleistì periorismì th S. Tìte, isqÔounta ex  :(1) To uposÔnolo TjQ tou T e�nai kleistì.(2) E�n h R e�nai (M;Co(Q))-epitrept , tìte TjCo(Q) = T n TjQ.(3) O periorismì Q e�nai (M;R)-pl rh periorismì .2.4.32 Prìtash. Upojètoume ìti o Q e�nai anoiktì periorismì th S kai ìti h oikogè-neia R e�nai (M;Co(Q))-epitrept . Tìte, isqÔoun ta ex  :(1) To uposÔnolo TjQ tou T e�nai anoiktì.(2) TjCo(Q) = T n TjQ.(3) O periorismì Q e�nai (M;R)-pl rh periorismì .2.4.33 Prìtash. Upojètoume ìti h oikogèneia R e�nai (M;Cl(Q))-epitrept . Tìte,TjCl(Q) = ClT(TjQ):2.4.34 Prìtash. Upojètoume ìti o Q e�nai (M;R)-pl rh periorismì kai ìti h oikogè-neia R e�nai (M; Int(Q))-epitrept  kai (M;Co(Int(Q)))-epitrept . Tìte,TjInt(Q) = IntT(TjQ):2.4.35 Prìtash. Upojètoume ìti o Q e�nai (M;R)-pl rh periorismì kai ìti h oiko-gèneia R e�nai (M;Cl(Q))-epitrept , (M; Int(Q))-epitrept , (M;Co(Int(Q)))-epitrept kai (M;Bd(Q))-epitrept . Tìte, TjBd(Q) = BdT(TjQ):

48 Kef�laio 22.4.36 Prìtash. Upojètoume ìti o Q e�nai (M;R)-pl rh periorismì kai ìti h oiko-gèneia R e�nai (M;Co(Q))-epitrept . Tìte, o Co(Q) e�nai (M;R)-pl rh periorismì kai TjCo(Q) = T n TjQ:

Kef�laio 3Koresmène kl�sei Sto kef�laio autì d�netai o orismì th koresmènh kl�sh q¸rwn. Mia kl�sh q¸rwnIP e�nai koresmènh ìtan gia k�je diktuwmènh oikogèneia S apì q¸rou pou an koun sthnIP o Periektikì Q¸ro T(M;R) an kei sthn kl�sh P gia {sqedìn ìla} ta sun-shm�diaM kai ti oikogèneie R. Sunep¸ , sthn per�ptwsh aut , oi Periektiko� Q¸roi T(M;R)ja e�nai kajoliko� gia thn kl�sh IP.Sto kef�laio autì ektì apì thn ènnoia th koresmènh kl�sh q¸rwn d�netai kai hènnoia th koresmènh kl�sh q¸rwn pou èqoun mia {dom }. Kl�sei q¸rwn me dom  e�naioi kl�sei uposunìlwn, oi kl�sei b�sewn kai oi kl�sei p-b�sewn. Gia autè ti kl�sei d�netai h ènnoia tou kajolikoÔ stoiqe�ou.Oi koresmène kl�sei èqoun fusik� thn idiìthta th kajolikìthta , dhlad  se k�jekoresmènh kl�sh up�rqei kajolikì stoiqe�o. Wstìso, oi koresmène kl�sei q¸rwn èqoun{k�ti parap�nw} apì thn Ôparxh kajolik¸n q¸rwn. Gia par�deigma, oi koresmène kl�sei èqoun thn idiìthta th tom  , dhlad  h tom  (to polÔ � to pl jo ) koresmènwn kl�sewne�nai ep�sh mia koresmènh kl�sh, parìlo pou h tom  kl�sewn pou èqoun kajolik� stoiqe�ampore� na mhn èqei kajolikì stoiqe�o.'Oloi oi orismo� kai oi prot�sei tou kefala�ou autoÔ br�skontai sto bibl�o [37℄.3.1 Koresmène kl�sei q¸rwnSthn enìthta aut  upojètoume ìti ìle oi kl�sei q¸rwn e�nai topologikè .3.1.1 Orismì . 'Estw S mia mh ken  diktuwmènh oikogèneia S shmademènwn topologik¸nq¸rwn kai M � ffUXÆ : Æ 2 �g : X 2 Sg;M+ � ffV XÆ : Æ 2 �g : X 2 Sg49

50 Kef�laio 3dÔo sun-shm�dia th S. Lème ìti to M e�nai sun-epèktash ( o-extension) tou M+, e�nup�rqei mia 1-1 sun�rthsh � : � ! � tètoia ¸ste gia k�je X 2 S kai gia k�je Æ 2 � naèqoume V XÆ = UX�(Æ). H sun�rthsh � kale�tai endeiktik  sun�rthsh th sun-epèktash   endeiktik  sun�rthsh apì to M+ sto M.3.1.2 Parat rhsh. Shmei¸noume ìti h sun�rthsh �, en gènei, den e�nai monadik .3.1.3 Parat rhsh. 'Estw S mia mh ken  diktuwmènh oikogèneia S shmademènwn topolo-gik¸n q¸rwn kai M0 � ffUXÆ : Æ 2 �g : X 2 Sg;M1 � ffV XÆ : Æ 2 �g : X 2 Sg;M2 � ffWXÆ : Æ 2 �g : X 2 Sgtr�a sun-shm�dia th S. E�n to M0 e�nai sun-epèktash tou M1 kai to M1 e�nai sun-epèktash tou M2, tìte to M0 e�nai sun-epèktash tou M2 (metabatik  idiìthta).Pr�gmati, epeid  to sun-shm�diM0 e�nai sun-epèktash tou M1, up�rqei mia 1-1 sun�r-thsh � : � ! � tètoia ¸ste gia k�je X 2 S kai gia k�je Æ 2 � , V XÆ = UX�(Æ). Ep�sh ,epeid  to sun-shm�diM1 e�nai sun-epèktash tou M2, up�rqei mia 1-1 sun�rthsh � : � ! �tètoia ¸ste gia k�je X 2 S kai gia k�je Æ 2 � , WXÆ = V X�(Æ). Jètoume = � Æ � : � ! � .H e�nai 1-1 kai epiplèon gia k�je X 2 S kai gia k�je Æ 2 � ,WXÆ = V X�(Æ) = UX�(�(Æ)) = UX(�Æ�)(Æ) = UX (Æ):3.1.4 Orismì . Mia kl�sh q¸rwn IP kale�tai koresmènh (saturated), e�n gia k�jediktuwmènh oikogèneia S apì q¸rou pou an koun sthn IP up�rqei èna sun-shm�diM+ th S ètsi ¸ste gia k�je sun-shm�di M th S, pou e�nai sun-epèktash tou M+, na up�rqeimia M-epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S tètoia ¸ste gia k�jeepitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, pou e�nai telik¸ leptìterh th R+, kai k�je L 2 C}(R) na èqoume ìti T(L) 2 IP.To sun-shm�diM+ kale�tai arqikì sun-shm�di (initial o-mark) th S (pou antistoi-qe� sthn kl�sh IP) kai h oikogèneia R+ kale�tai arqik  oikogèneia th S (pou antistoiqe�sto sun-shm�di M kai sthn kl�sh IP). H ken  kl�sh q¸rwn jewre�tai koresmènh.3.1.5 Prìtash. 'Estw S mia mh ken  diktuwmènh oikogèneia S shmademènwn topologik¸nq¸rwn.(1) Upojètoume ìti to polÔ � to pl jo epitreptè oikogèneie apì sqèsei isodunam�a ep� th S e�nai dosmène . Tìte, up�rqei mia epitrept  oikogèneia apì sqèsei isodunam�a ep� th S, h opo�a e�nai telik¸ leptìterh apì k�je dosmènh oikogèneia.

Koresmène kl�sei 51(2) Upojètoume ìti to polÔ � to pl jo sun-shm�dia th S èqoun doje�. Tìte, up�rqeièna sun-shm�di th S, to opo�o e�nai sun-epèktash apì k�je dosmèno sun-shm�di.3.1.6 Prìtash. H tom  to polÔ � to pl jo koresmènwn kl�sewn q¸rwn e�nai ep�sh mia koresmènh kl�sh q¸rwn.'Estw IP mia koresmènh kl�sh q¸rwn. Epeid  h IP e�nai topologik , up�rqei mia oikogè-neia S apì q¸rou pou an koun sthn IP ètsi ¸ste k�je q¸ro pou an kei sthn IP na e�naiomoiìmorfo me èna q¸ro pou an kei sthn S. Ep�sh , epeid  h IP e�nai koresmènh up�rqeièna sun-shm�di M th S kai mia M-epitrept  oikogèneia R apì sqèsei isodunam�a ep�th S ¸ste T(M;R) = T 2 IP.O Periektikì Q¸ro T th oikogèneia S pou antistoiqe� sto sun-shm�di M kai sthnoikogèneia R e�nai kajolikì gia thn kl�sh IP. Pr�gmati, epeid  T 2 IP, arke� na de�xoumeìti k�je q¸ro Z 2 IP perièqetai topologik� sto q¸ro T. 'Estw Z 2 IP kai X to stoiqe�oth S pou e�nai omoiìmorfo me to Z. Sumbol�zoume me h ènan omoiomorfismì tou Z ep� touX kai jewroÔme th fusik  emfÔteush eXT tou X sto T . H apeikìnish e � eXT Æ h e�nai miaemfÔteush tou Z sto T . Z h //e @

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@XeXT��T3.1.7 Prìtash. K�je mh ken  koresmènh kl�sh q¸rwn èqei kajolikì q¸ro.3.1.8 Parade�gmata koresmènwn kl�sewn q¸rwn.(1) H kl�sh ìlwn twn T0-q¸rwn me b�ro � � .(2) H kl�sh ìlwn twn kanonik¸n q¸rwn me b�ro � � .(3) H kl�sh ìlwn twn Ty hono� q¸rwn me b�ro � � .3.2 Koresmène kl�sei uposunìlwn3.2.1 Orismì . Me ton ìro kl�sh uposunìlwn ennooÔme mia kl�sh IP pou apotele�taiapì zeÔgh (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X. Mia kl�sh uposunìlwnIP kale�tai topologik , e�n gia k�je omoiomorfismì h : X ! Y h sunj kh (Q;X) 2 IPsunep�getai ìti (h(Q); Y ) 2 IP.Se ì,ti akolouje� upojètoume ìti ìle oi kl�sei uposunìlwn e�nai topologikè .

52 Kef�laio 33.2.2 Orismì . 'Estw IP mia kl�sh uposunìlwn. O periorismì Q = fQX : X 2 Sg mia diktuwmènh oikogèneia S topologik¸n q¸rwn kale�tai IP-periorismì (IP-restri tion),e�n (QX ; X) 2 IP gia k�je X 2 S.3.2.3 Orismì . Mia mh ken  kl�sh uposunìlwn IP kale�tai koresmènh (saturated),e�n gia k�je diktuwmènh oikogèneia S apì q¸rou kai gia k�je IP-periorismì Q th Sup�rqei èna sun-shm�di M+ th S ètsi ¸ste gia k�je sun-shm�di M th S, pou e�nai sun-epèktash touM+, na up�rqei mia (M;Q)-epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S tètoia ¸ste gia k�je epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, pou e�nai telik¸ leptìterh th R+, kai k�je H;L 2 C}(R) me H � L na èqoume ìti(T(HjQ);T(L)) 2 IP.To sun-shm�diM+ kale�tai arqikì sun-shm�di (initial o-mark) th S (pou antistoi-qe� ston periorismì Q kai sthn kl�sh IP) kai h oikogèneia R+ kale�tai arqik  oikogèneiath S (pou antistoiqe� sto sun-shm�di M, ston periorismì Q kai sthn kl�sh IP).3.2.4 Orismì . 'Estw IP kl�sh uposunìlwn. 'Ena zeÔgo (QT ; T ), ìpou QT e�nai ènauposÔnolo tou q¸rou T , kale�tai kajolikì stoiqe�o (universal element) gia thn kl�shIP, ìtan:(1) (QT ; T ) 2 IP.(2) Gia k�je (QZ ; Z) 2 IP, up�rqei topologik  emfÔteush e : Z ! T tètoia ¸ste e(QZ) �QT .3.2.5 Prìtash. H tom  to polÔ � to pl jo koresmènwn kl�sewn uposunìlwn e�naiep�sh mia koresmènh kl�sh uposunìlwn.3.2.6 Prìtash. K�je mh ken  koresmènh kl�sh uposunìlwn èqei kajolikì stoiqe�o.3.2.7 Parade�gmata koresmènwn kl�sewn uposunìlwn.(1) H kl�sh IP(Cl) uposunìlwn pou apotele�tai apì ìla ta zeÔgh (Q;X), ìpou Q e�naièna kleistì uposÔnolo enì q¸rou X.(2) H kl�sh IP(Op) uposunìlwn pou apotele�tai apì ìla ta zeÔgh (Q;X), ìpou Q e�naièna anoiktì uposÔnolo enì q¸rou X.(3) H kl�sh IP(n:dense) uposunìlwn pou apotele�tai apì ìla ta zeÔgh (Q;X), ìpou Qe�nai èna poujen� puknì uposÔnolo enì q¸rou X.3.2.8 Orismì . 'Ena periorismì Q mia diktuwmènh oikogèneia q¸rwn S kale�tai pl -rh ( omplete), e�n up�rqei èna sun-shm�di M th S kai mia (M;Q)-epitrept  oikogèneiaR apì sqèsei isodunam�a ep� th S ètsi ¸ste o Q na e�nai (M;R)-pl rh periorismì .

Koresmène kl�sei 533.2.9 Orismì . Mia kl�sh uposunìlwn IP kale�tai pl rh ( omplete), e�n gia k�jediktuwmènh oikogèneia S apì q¸rou k�je IP-periorismì th S e�nai pl rh .3.2.10 Parade�gmata pl rwn kl�sewn uposunìlwn.(1) H kl�sh IP(Cl).(2) H kl�sh IP(Op) .(3) H kl�sh IP(Cl) \ IP(n:dense).3.3 Koresmène kl�sei b�sewn3.3.1 Orismì . Me ton ìro kl�sh b�sewn ennooÔme mia kl�sh IP pou apotele�tai apìzeÔgh (B;X), ìpou B e�nai mia b�sh enì q¸rou X me jBj � � . Mia kl�sh b�sewn IPkale�tai topologik , e�n gia k�je omoiomorfismì h : X ! Y h sunj kh (B;X) 2 IPsunep�getai ìti (fh(V ) : V 2 Bg; Y ) 2 IP.Se ì,ti akolouje� upojètoume ìti ìle oi kl�sei b�sewn e�nai topologikè .3.3.2 Orismì . 'Estw IP kl�sh b�sewn. Mia b�sh B enì q¸rou X kale�tai IP-b�sh(IP-base), e�n (B;X) 2 IP.3.3.3 Orismì . 'Estw S mia diktuwmènh oikogèneia q¸rwn.(1) 'Ena S-diktuwmèno sÔnolo B � fBX : X 2 Sg apì oikogèneie kale�tai sun-b�shth S ( o-base for S), e�n h BX e�nai mia b�sh tou X me jBX j � � .(2) 'Ena S-diktuwmèno sÔnolo N � fNX : X 2 Sg apì � -diktuwmène oikogèneie kale�taisun-diktÔwsh ( o-indi ation) mia sun-b�sh B � fBX : X 2 Sg th S, e�n to NX e�naimia diktÔwsh tou BX gia k�je X 2 S.(3) Mia sun-b�sh B � fBX : X 2 Sg th S kale�tai IP-sun-b�sh (IP- o-base), e�n giak�je X 2 S h oikogèneia BX e�nai mia IP-b�sh tou X.3.3.4 Orismì . Mia mh ken  kl�sh b�sewn IP kale�tai koresmènh (saturated), e�ngia k�je diktuwmènh oikogèneia S apì q¸rou , gia k�je IP-sun-b�sh B th S kai giak�je sun-diktÔwsh N th B up�rqei èna sun-shm�di M+ th S, sun-epèktash tou N ètsi¸ste gia k�je sun-shm�di M th S, pou e�nai sun-epèktash tou M+, na up�rqei mia M-epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S tètoia ¸ste gia k�je epitrept oikogèneia R apì sqèsei isodunam�a ep� th S, pou e�nai telik¸ leptìterh th R+, kaik�je L 2 C}(R) na èqoume ìti (BL};�(�);T(L)) 2 IP, ìpou � e�nai mia endeiktik  sun�rthshapì to N sto M.

54 Kef�laio 3To sun-shm�diM+ kale�tai arqikì sun-shm�di (initial o-mark) th S (pou antistoi-qe� sth sun-diktÔwsh N th B kai sthn kl�sh IP) kai h oikogèneia R+ kale�tai arqik oikogèneia th S (pou antistoiqe� sto sun-shm�di M, sth sun-diktÔwsh N th B kaisthn kl�sh IP).3.3.5 Orismì . 'Estw IP kl�sh b�sewn. 'Ena zeÔgo (BT ; T ), ìpou BT e�nai mia b�shtou q¸rou T me jBT j � � , kale�tai kajolikì stoiqe�o (universal element) gia thn kl�shIP, ìtan:(1) (BT ; T ) 2 IP.(2) Gia k�je (BX ; X) 2 IP, up�rqei topologik  emfÔteush e : X ! T tètoia ¸steBX = fe�1(V ) : V 2 BTg.3.3.6 Prìtash. H tom  to polÔ � to pl jo koresmènwn kl�sewn b�sewn e�nai ep�sh mia koresmènh kl�sh b�sewn.3.3.7 Prìtash. K�je mh ken  koresmènh kl�sh b�sewn èqei kajolikì stoiqe�o.3.3.8 Prìtash. 'Estw IP koresmènh kl�sh b�sewn kai IE koresmènh kl�sh q¸rwn. Hkl�sh f(B;X) 2 IP : X 2 IEge�nai koresmènh kl�sh b�sewn.3.4 Koresmène kl�sei p-b�sewn3.4.1 Orismì . 'Estw Q èna uposÔnolo enì q¸rou X. Mia oikogèneia B apì anoikt�uposÔnola tou X (sumperilambanomènwn twn X kai ;) kale�tai p-b�sh gia to Q stoX, e�n to sÔnolo fQ \ U : U 2 Bg e�nai mia b�sh tou upoq¸rou Q. Mia p-b�sh gia toQ sto X kale�tai pos-b�sh, e�n gia k�je x 2 Q kai gia k�je anoikt  perioq  U tou xsto X up�rqei èna stoiqe�o V th B ètsi ¸ste x 2 V � U . Mia p-b�sh gia to Q sto Xkale�tai ps-b�sh, e�n h oikogèneia B e�nai mia b�sh tou q¸rou X.3.4.2 Orismì . Me ton ìro kl�sh p-b�sewn ennooÔme mia kl�sh IP pou apotele�taiapì tri�de (Q;B;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X kai B e�nai mia p-b�shgia to Q sto X me jBj � � . Mia kl�sh p-b�sewn IP kale�tai topologik , e�n gia k�jeomoiomorfismì h : X ! Y h sunj kh (Q;B;X) 2 IP sunep�getai ìti(h(Q); fh(V ) : V 2 Bg; Y ) 2 IP:

Koresmène kl�sei 55Me ton ìro kl�sh pos-b�sewn (ant�stoiqa, kl�sh ps-b�sewn) ennooÔme mia kl�shIP p-b�sewn pou apotele�tai apì tri�de (Q;B;X), ìpou B e�nai mia pos-b�sh gia to Qsto X (ant�stoiqa, mia ps-b�sh gia to Q sto X).Se ì,ti akolouje� upojètoume ìti ìle oi kl�sei p-b�sewn e�nai topologikè .3.4.3 Orismì . 'Estw IP kl�sh p-b�sewn. Mia p-b�sh B gia èna uposÔnolo Q enì q¸rou X kale�tai IP-p-b�sh (IP-p-base), e�n (Q;B;X) 2 IP.3.4.4 Orismì . 'Estw S mia oikogèneia q¸rwn kai Q � fQX : X 2 Sg èna periorismì th S.(1) 'Ena S-diktuwmèno sÔnolo B � fBX : X 2 Sg apì oikogèneie kale�tai sun-p-b�shgia ton Q sto S ( o-base for Q in S) (ant�stoiqa, sun-pos-b�sh   sun-ps-b�sh giaton Q sto S), e�n h BX e�nai mia p-b�sh (ant�stoiqa, mia pos-b�sh   ps-b�sh) gia to QXsto X.(2) Mia sun-p-b�sh B � fBX : X 2 Sg gia ton Q sto S kale�tai IP-sun-p-b�sh (IP-p- o-base), e�n gia k�je X 2 S h oikogèneia BX e�nai mia IP-p-b�sh gia to QX sto X.3.4.5 Orismì . Mia mh ken  kl�sh p-b�sewn IP kale�tai koresmènh (saturated), e�ngia k�je diktuwmènh oikogèneia S apì q¸rou , gia k�je periorismì Q th S, gia k�jeIP-sun-p-b�sh B gia ton Q sto S kai gia k�je sun-diktÔwsh N th B up�rqei èna sun-shm�di M+ th S, sun-epèktash tou N ètsi ¸ste gia k�je sun-shm�di M th S, poue�nai sun-epèktash tou M+, na up�rqei mia (M;Q)-epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S tètoia ¸ste gia k�je epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, pou e�nai telik¸ leptìterh th R+, kai k�je L;H;E 2 C}(R) me L � H � Ena èqoume ìti (T(EjQ);BH};�(�);T(L)) 2 IP, ìpou � e�nai mia endeiktik  sun�rthsh apì toN sto M.To sun-shm�diM+ kale�tai arqikì sun-shm�di (initial o-mark) th S (pou antistoi-qe� ston periorismì Q, sth sun-diktÔwshN th B kai sthn kl�sh IP) kai h oikogèneia R+kale�tai arqik  oikogèneia th S (pou antistoiqe� sto sun-shm�di M, ston periorismìQ, sth sun-diktÔwsh N th B kai sthn kl�sh IP).3.4.6 Orismì . 'Estw IP kl�sh p-b�sewn. 'Ena zeÔgo (QT ; BT ; T ), ìpou QT e�nai ènauposÔnolo tou q¸rou X kai BT e�nai mia p-b�sh gia to Q sto X me jBT j � � , kale�taikajolikì stoiqe�o (universal element) gia thn kl�sh IP, ìtan:(1) (QT ; BT ; T ) 2 IP.

56 Kef�laio 3(2) Gia k�je (QZ ; BZ ; Z) 2 IP, up�rqei topologik  emfÔteush e : Z ! T tètoia ¸stee(QZ) � QT kai BZ = fe�1(V ) : V 2 BTg.3.4.7 Prìtash. H tom  to polÔ � to pl jo koresmènwn kl�sewn p-b�sewn e�nai ep�sh mia koresmènh kl�sh p-b�sewn.3.4.8 Prìtash. K�je mh ken  koresmènh kl�sh p-b�sewn èqei kajolikì stoiqe�o.3.4.9 Prìtash. 'Estw IP koresmènh kl�sh p-b�sewn, IF koresmènh kl�sh uposunìlwnkai IE koresmènh kl�sh q¸rwn. Oi kl�sei f(Q;B;X) 2 IP : (Q;X) 2 IFg,f(Q;B;X) 2 IP : Q 2 IEg kaif(Q;B;X) 2 IP : X 2 IEge�nai koresmène kl�sei p-b�sewn.

MEROS B

Kef�laio 4Diast�sei tou tÔpou IndSthn ergas�a [56℄ or�sjhkan dÔo diast�sei dm kai Dm sthn kl�sh ìlwn twn Haus-dor� q¸rwn. H di�stash Dm den èqei thn idiìthta th kajolikìthta sthn kl�sh ìlwntwn diaqwr�simwn metrikopoi simwn q¸rwn epeid  h oikogèneia ìlwn twn diaqwr�simwn me-trikopoi simwn q¸rwn X me Dm(X) � 0 sump�ptei me thn oikogèneia ìlwn twn totallydis onne ted q¸rwn sthn opo�a den up�rqoun kajolik� stoiqe�a (blèpe [65℄).Sto kef�laio autì tropopoioÔntai oi diast�sei dm kai Dm me skopì na orisjoÔnkainoÔrgie diast�sei pou èqoun thn idiìthta th kajolikìthta . Autè oi kainoÔrgie diast�sei sumbol�zontai me dmIK;IBIE;� kai DmIK;IBIE;� , ìpou IE e�nai mia kl�sh q¸rwn, IK e�naimia kl�sh uposunìlwn kai IB e�nai mia kl�sh b�sewn. Sto kef�laio autì melet¸ntai oidiast�sei dmIK;IBIE;� kai DmIK;IBIE;� kai sugkr�nontai me �lle gnwstè diast�sei . Eidikìtera,apodeiknÔetai ìti e�n oi kl�sei IK; IB kai IE e�nai koresmène , tìte gia mia dosmènh ko-resmènh kl�sh IP q¸rwn kai gia èna fusikì arijmì � h kl�sh ìlwn twn q¸rwn X pouan koun sthn kl�sh IP me dmIK;IBIE;� (X) � � kai h kl�sh ìlwn twn q¸rwn X pou an -koun sthn kl�sh IP me DmIK;IBIE;� (X) � � èqoun kajolik� stoiqe�a. Ta apotelèsmata toukefala�ou autoÔ e�nai ìla prwtìtupa.Se ì,ti akolouje� me th lèxh q¸ro ja ennooÔme ènan T0-q¸ro me b�ro � � .4.1 Oi diast�sei dm kai DmSthn ergas�a [56℄ dÔo diast�sei dm kaiDm or�sjhkan kai melet jhkan. Sthn ergas�a[6℄ oi diast�sei autè epekt�jhkan sthn kl�sh O ìlwn twn diataktik¸n arijm¸n kaisumbol�sthkan me trdm kai trDm. Parak�tw d�nontai oi orismo� aut¸n twn epekt�sewnqrhsimopoi¸nta ìmw tou arqikoÔ sumbolismoÔ dm kai Dm.4.1.1 Orismì . JewroÔme ti sunart sei dm kai Dm me ped�o orismoÔ thn kl�sh ìlwntwn q¸rwn kai ped�o tim¸n thn kl�shO[f�1;1g pou ikanopoioÔn ti parak�tw sunj ke :59

60 Kef�laio 4(1) dm(X) = Dm(X) = �1 e�n kai mìno e�n X = ;.(2) Dm(X) � �, ìpou � 2 O, e�n gia k�je zeÔgo diakekrimènwn shme�wn x kai y tou Xup�rqei uposÔnolo L tou X pou diaqwr�zei ta sÔnola fxg kai fyg me dm(L) < �.(3) dm(X) � �, ìpou � 2 O, e�n X = [fQXi : i 2 !g, ìpou to QXi e�nai kleistìuposÔnolo tou X me Dm(QXi ) � �, i 2 !.Epomènw , dm(X) =1 (ant�stoiqa, Dm(X) =1) an kai mìno an h anisìthta dm(X) � �(ant�stoiqa, Dm(X) � �) den e�nai alhj  gia k�je � 2 O.Sti ergas�e [56℄ kai [6℄ oi diast�sei dm kai Dm or�sjhkan sthn kl�sh ìlwn twnHausdor� q¸rwn. Apì ton parap�nw orismì prokÔptei ìti e�n èna q¸ro X den e�naiHausdor�, tìte dm(X) = Dm(X) =1.4.1.2 Parat rhsh. E�nai eÔkolo na de� kane� ìti Dm(X) = 0 e�n kai mìno e�n oq¸ro X e�nai totally dis onne ted. E�nai gnwstì ìti sthn kl�sh ìlwn twn diaqwr�simwnmetrikopoi simwn totally dis onne ted q¸rwn den up�rqoun kajolik� stoiqe�a (blèpe [65℄).4.2 Oi diast�sei dmIK;IBIE;� kai DmIK;IBIE;�Se ì,ti akolouje� me � sumbol�zoume èna stajerì plhj�rijmo tètoio ¸ste ! � � � � .4.2.1 Orismì . Mia kl�sh q¸rwn IE kale�tai IB-hereditary-separated, ìpou IB e�naimia kl�sh b�sewn, e�n gia k�je X 2 IE up�rqei mia IB-b�sh BX � fUÆ : Æ 2 �g tou Xètsi ¸ste gia k�je dÔo stoiqe�a UÆ1 kai UÆ2 th BX me Cl(UÆ1) \ Cl(UÆ2) = ; na up�rqeièna upìqwro L tou X pou an kei sthn kl�sh IE kai diaqwr�zei ta sÔnola Cl(UÆ1) kaiCl(UÆ2).Shmei¸noume ìti e�n h kl�sh IE e�nai IB-hereditary-separated, tìte ; 2 IE, to opo�oprokÔptei apì to gegonì ìti to kenì sÔnolo e�nai to monadikì uposÔnolo tou X poudiaqwr�zei ta stoiqe�a ; kai X th BX .4.2.2 Orismì . 'Estw IB mia kl�sh b�sewn, IE mia IB-hereditary-separated kl�sh q¸rwnkai IK mia kl�sh uposunìlwn me (X;X) 2 IK gia k�je q¸ro X. JewroÔme ti sunart sei dmIK;IBIE;� kai DmIK;IBIE;� me ped�o orismoÔ thn kl�sh ìlwn twn q¸rwn kai ped�o tim¸n thn kl�shO [ f�1;1g pou ikanopoioÔn ti parak�tw sunj ke :(1) dmIK;IBIE;� (X) = DmIK;IBIE;� (X) = �1 e�n kai mìno e�n X 2 IE.(2) DmIK;IBIE;� (X) � �, ìpou � 2 O, e�n up�rqei mia IB-b�sh BX � fUÆ : Æ 2 �g tou X ètsi¸ste gia k�je dÔo stoiqe�a UÆ1 kai UÆ2 th BX me Cl(UÆ1) \ Cl(UÆ2) = ; na up�rqei èna upìqwro L tou X pou diaqwr�zei ta sÔnola Cl(UÆ1) kai Cl(UÆ2) me dmIK;IBIE;� (L) < �.

Diast�sei tou tÔpou Ind 61(3) dmIK;IBIE;� (X) � �, ìpou � 2 O, e�n X = [fQXi : i 2 �g ¸ste (a) to uposÔnolo QXi touX e�nai kleistì, (b) (QXi ; X) 2 IK kai (g) DmIK;IBIE;� (QXi ) � �, i 2 �.Epomènw , dmIK;IBIE;� (X) = 1 (ant�stoiqa, DmIK;IBIE;� (X) = 1) e�n kai mìno e�n h anisìthtadmIK;IBIE;� (X) � � (ant�stoiqa, DmIK;IBIE;� (X) � �) den e�nai alhj  gia k�je � 2 O.4.2.3 Parat rhsh. (1) Oi sunart sei dmIK;IBIE;� kai DmIK;IBIE;� e�nai kal� orismène . Pr�g-mati, arke� na apode�xoume ìti e�n gia èna q¸ro X èqoumeDmIK;IBIE;� (X) = dmIK;IBIE;� (X) = �1;tìte DmIK;IBIE;� (X) � 0 kai dmIK;IBIE;� (X) � 0:Ef> ìson X 2 IE, h sqèsh DmIK;IBIE;� (X) � 0 prokÔptei �mesa apì to gegonì ìti h kl�shIE e�nai IB-hereditary-separated. H sqèsh dmIK;IBIE;� (X) � 0 prokÔptei apì to gegonì ìtiX = [fQXi : i 2 �g, ìpou QXi = X, (X;X) 2 IK kai DmIK;IBIE;� (X) = �1 � 0.(2) Se ì,ti akolouje� ìtan jewroÔme ti diast�sei dmIK;IBIE;� kai DmIK;IBIE;� , ja upojètoume ìtiIB e�nai mia kl�sh b�sewn, IE e�nai mia IB-hereditary-separated kl�sh q¸rwn kai IK e�naimia kl�sh uposunìlwn me (X;X) 2 IK gia k�je q¸ro X. Sthn per�ptwsh, ìpou IE = f;gant� gia dmIK;IBIE;� kai DmIK;IBIE;� ja gr�foume dmIK;IB� kai DmIK;IB� , ant�stoiqa.4.3 Sqèsei metaxÔ twn dmIK;IBIE;� , DmIK;IBIE;� kai �llwndiast�sewn.4.3.1 Prìtash. Gia k�je q¸ro X èqoume(4:1) dmIK;IBIE;� (X) � DmIK;IBIE;� (X):Apìdeixh. 'Estw DmIK;IBIE;� (X) = � 2 f�1;1g[O. E�n � = �1   � =1, tìte profan¸ h anisìthta (4.1) isqÔei. Upojètoume ìti � 2 O. Tìte,X = [fQXi : i 2 �g;ìpou QXi = X, i 2 �. Ef> ìson (QXi ; X) = (X;X) 2 IKkai DmIK;IBIE;� (QXi ) = DmIK;IBIE;� (X) � �;

62 Kef�laio 4apì th sunj kh (3) tou OrismoÔ 4.2.2 prokÔptei ìti dmIK;IBIE;� (X) � �. �4.3.2 Prìtash. Gia k�je q¸ro X èqoumeDmIK;IBIE;� (X) 2 f�1;1g[ �+kai sunep¸ dmIK;IBIE;� (X) 2 f�1;1g[ �+:Apìdeixh. Upojètoume ìti h prìtash den e�nai alhj  . Ja katal xoume se �topo. 'Estw� to el�qisto stoiqe�o tou O n �+ tètoio ¸ste up�rqei èna q¸ro X me DmIK;IBIE;� (X) = �.JewroÔme mia IB-b�sh BX � fUÆ : Æ 2 �g tou X pou ikanopoie� th sunj kh (2) touOrismoÔ 4.2.2 kai sumbl�zoume me P to sÔnolo twn zeug¸n (Æ1; Æ2) 2 � � � meCl(UÆ1) \ Cl(UÆ2) = ;:Gia k�je (Æ1; Æ2) 2 P èstw L(Æ1; Æ2) èna uposÔnolo touX pou diaqwr�zei ta sÔnola Cl(UÆ1)kai Cl(UÆ2) me dmIK;IBIE;� (L(Æ1; Æ2)) = �(Æ1; Æ2) < �:Pr¸ta upojètoume ìti �(Æ1; Æ2) < �+ gia k�je (Æ1; Æ2) 2 P . Ef> ìson jP j � � , up�rqeièna diataktikì arijmì � 2 �+ tètoio ¸ste �(Æ1; Æ2) < � gia k�je (Æ1, Æ2) 2 P . Tìte,dmIK;IBIE;� (L(Æ1; Æ2)) < � kai, apì th sunj kh (2) tou OrismoÔ 4.2.2, DmIK;IBIE;� (X) � �, �topo.T¸ra, upojètoume ìti up�rqei (Æ1; Æ2) 2 P tètoio ¸ste �+ � �(Æ1; Æ2). Ef> ìsondmIK;IBIE;� (L(Æ1; Æ2)) = �(Æ1; Æ2), up�rqoun kleist� uposÔnola QL(Æ1;Æ2)i tou L(Æ1; Æ2), i 2 �,ètsi ¸ste:(a) L(Æ1; Æ2) = [fQL(Æ1 ;Æ2)i : i 2 �g,(b) (QL(Æ1;Æ2)i ; L(Æ1; Æ2)) 2 IK kai(g) DmIK;IBIE;� (QL(Æ1;Æ2)i ) = �i � �(Æ1; Æ2) < �.E�n �i < �+ gia k�je i 2 �, tìte up�rqei èna diataktikì arijmì � 2 �+ tètoio ¸ste�i � �, pou shma�nei ìti DmIK;IBIE;� (QL(Æ1;Æ2)i ) � �. Sunep¸ ,dmIK;IBIE;� (L(Æ1; Æ2)) � � < �+ � �(Æ1; Æ2);�topo. 'Ara, up�rqei i 2 � tètoio ¸ste�+ � DmIK;IBIE;� (QL(Æ1;Æ2)i ) < �:

Diast�sei tou tÔpou Ind 63H teleuta�a sqèsh antif�skei thn epilog  tou diataktikoÔ arijmoÔ �. Se ìle ti peri-pt¸sei katal xame se �topo. Sunep¸ , h prìtash e�nai alhj  . �4.3.3 Prìtash. 'Estw X kanonikì q¸ro . Tìte,dm(X) � dmIK;IB! (X)kai Dm(X) � DmIK;IB! (X):Apìdeixh. Ja apode�xoume thn prìtash me epagwg . An dmIK;IB! (X) = DmIK;IB! (X) = �1,tìte X = ; kai sunep¸ , apì th sunj kh (1) tou OrismoÔ 4.1.1, dm(X) = Dm(X) = �1.Upojètoume ìti h anisìthta Dm(X) � DmIK;IB! (X) e�nai alhj  gia k�je kanonikì q¸roX meDmIK;IB! (X) < � kai h anisìthta dm(X) � dmIK;IB! (X) e�nai alhj  gia k�je kanonikìq¸ro X me dmIK;IB! (X) < �, ìpou � 2 O.'Estw X èna kanonikì q¸ro me DmIK;IB! (X) = �. ApodeiknÔoume ìti Dm(X) � �.'Estw x kai y dÔo diakekrimèna shme�a tou X. Up�rqei mia IB-b�sh BX � fUÆ : Æ 2 �g touX ètsi ¸ste gia k�je dÔo stoiqe�a UÆ1 kai UÆ2 th BX me Cl(UÆ1)\Cl(UÆ2) = ; na up�rqeièna upìqwro L tou X pou diaqwr�zei ta sÔnola Cl(UÆ1) kai Cl(UÆ2) me dmIK;IB! (L) < �.Ef> ìson o q¸ro X e�nai kanonikì up�rqoun UÆ1 , UÆ2 2 BX ètsi ¸ste: x 2 Cl(UÆ1),y 2 Cl(UÆ2) kai Cl(UÆ1)\Cl(UÆ2) = ;. Sunep¸ , up�rqei upìqwro L tou X pou diaqwr�zeita sÔnola Cl(UÆ1) kai Cl(UÆ2) me dmIK;IB! (L) < �. Profan¸ , o L diaqwr�zei ta monosÔnolafxg kai fyg. Epiplèon apì thn upìjesh th epagwg  dm(L) � dmIK;IB! (L) < �:Epomènw , apì th sunj kh (2) tou OrismoÔ 4.1.1, Dm(X) � �.'Estw t¸ra X kanonikì q¸ro me dmIK;IB! (X) = �. ApodeiknÔoume ìti dm(X) � �.'Eqoume X = [fQXi : i 2 !g;ìpou:(a) to uposÔnolo QXi tou X e�nai kleistì,(b) (QXi ; X) 2 IK kai(g) DmIK;IB! (QXi ) � �.Apì to pr¸to mèro th apìdeixh prokÔptei ìti Dm(QXi ) � �, i 2 !. 'Ara, apì thsunj kh (3) tou OrismoÔ 4.1.1, dm(X) � �.Tèlo , h prìtash e�nai profan  e�n dmIK;IB! (X) = DmIK;IB! (X) =1. �

64 Kef�laio 44.3.4 Prìtash. 'Estw X fusikì q¸ro me DmIK;IB� (X) 6=1. Tìte,DmIK;IB� (X) � Ind(X):Apìdeixh. 'Estw Ind(X) = � 2 f�1;1g[O. Ja apode�xoume thn prìtash me epagwg sto �. H prìtash e�nai profan  e�n � = �1   � = 1. 'Estw � 2 O kai upojètoumeìti h prìtash e�nai alhj  gia k�je fusikì q¸ro X me Ind(X) < �. ApodeiknÔoume thnprìtash gia èna fusikì q¸ro X me Ind(X) = �.'Estw BX � fUÆ : Æ 2 �g mia tuqa�a IB-b�sh tou X. H Ôparxh mia tètoia b�sh èpetai apì th sunj kh DmIK;IB� (X) 6=1. 'Estw UÆ1 , UÆ2 2 BX me Cl(UÆ1) \ Cl(UÆ2) = ;.Ef> ìson Ind(X) = �, up�rqei kleistì upìqwro L tou X pou diaqwr�zei ta sÔnolaCl(UÆ1) kai Cl(UÆ2) tètoio ¸ste Ind(L) < � (blèpe Parat rhsh 1.2.8). Apì thn upìjeshth epagwg  , DmIK;IB� (L) � Ind(L) < �:Apì thn Prìtash 4.3.1, èqoume dmIK;IB� (L) � DmIK;IB� (L):Autì shma�nei ìti DmIK;IB� (X) � �: �4.3.5 Pìrisma. 'Estw X fusikì q¸ro me DmIK;IB� (X) 6=1. Tìte,dmIK;IB� (X) � Ind(X):Apìdeixh. ProkÔptei �mesa apì ti Prot�sei 4.3.1 kai 4.3.4. �4.3.6 Prìtash. 'Estw X sumpag  Hausdor� q¸ro me DmIK;IB� (X) 6= 1. Tìte, oisunj ke DmIK;IB� (X) = 0 kai Ind(X) = 0 e�nai isodÔname .Apìdeixh. Apì thn Prìtash 4.3.4, arke� na apode�xoume ìti e�n DmIK;IB� (X) = 0, tìteInd(X) = 0. 'Estw DmIK;IB� (X) = 0 kai BX � fUÆ : Æ 2 �g mia IB-b�sh tou X ètsi ¸stegia k�je dÔo stoiqe�a UÆ1 kai UÆ2 th BX me Cl(UÆ1) \ Cl(UÆ2) = ; to kenì sÔnolo nadiaqwr�zei ta sÔnola Cl(UÆ1) kai Cl(UÆ2). ApodeiknÔoume ìti to kenì sÔnolo diaqwr�zeik�je dÔo, xèna metaxÔ tou , kleist� uposÔnola tou X (blèpe Parat rhsh 1.2.8).'Estw A;B zeÔgo xènwn kleist¸n uposunìlwn tou X. Ef> ìson o q¸ro X e�naifusikì , up�rqoun dÔo anoikt� uposÔnola U kai V tou X ètsi ¸ste: A � U � Cl(U),B � V � Cl(V ) kai Cl(U) \ Cl(V ) = ;. 'EstwU = [fUÆ : Æ 2 � � �g kai V = [fUÆ : Æ 2 � � �g:

Diast�sei tou tÔpou Ind 65Ef> ìson ta uposÔnola A kai B tou X e�nai kleist� kai o q¸ro X e�nai sumpag  , ta Akai B e�nai sumpag . Epomènw , up�rqoun s, t 2 F ètsi ¸steA � [fUÆ : Æ 2 sg � [fCl(UÆ) : Æ 2 sgkai B � [fUÆ : Æ 2 tg � [fCl(UÆ) : Æ 2 tg:Profan¸ , ([fCl(UÆ) : Æ 2 sg) \ ([fCl(UÆ) : Æ 2 tg) = ; kai �ra Cl(UÆ1) \ Cl(UÆ2) = ;gia k�je Æ1 2 s kai Æ2 2 t. Sunep¸ , up�rqoun anoikt� kai sugqrìnw kleist� uposÔnolaU Æ2Æ1 , Æ1 2 s, Æ2 2 t tou X ètsi ¸ste: Cl(UÆ1) � U Æ2Æ1 kai Cl(UÆ2) � X n U Æ2Æ1 gia k�je Æ1 2 skai Æ2 2 t.Jètoume W = \f[fU Æ2Æ1 : Æ1 2 sg : Æ2 2 tg:Tìte, A � W kai B � X n W . Ef> ìson ta uposÔnola U Æ2Æ1 , Æ1 2 s, Æ2 2 t, tou Xe�nai anoikt� kai sugqrìnw kleist�, to sÔnolo W e�nai anoiktì kai sugqrìnw kleistì.Sunep¸ , to kenì sÔnolo diaqwr�zei ta A kai B, pou shma�nei ìti Ind(X) = 0. �4.3.7 Prìtash. 'Estw X sumpag  Hausdor� q¸ro me DmIK;IB! (X) 6= 1. Tìte, oisunj ke dmIK;IB! (X) = 0 kai Ind(X) = 0 e�nai isodÔname .Apìdeixh. E�n Ind(X) = 0, tìte apì to Pìrisma 4.3.5,dmIK;IB! (X) = 0:Antistrìfw , èstw dmIK;IB! (X) = 0. Tìte,X = [fQXi : i 2 !gètsi ¸ste:(a) to uposÔnolo QXi tou X e�nai kleistì,(b) (QXi ; X) 2 IK kai(g) DmIK;IB! (QXi ) � 0.Ef> ìson o upìqwro QXi tou X e�nai sumpag  kai DmIK;IB! (QXi ) � 0, apì thn Prìtash4.3.6, Ind(QXi ) � 0. Sunep¸ , apì to Je¸rhma 1.2.13, Ind(X) = 0. �4.3.8 Pìrisma. 'Estw X sumpag  Hausdor� q¸ro me DmIK;IB! (X) 6= 1. Tìte, oiparak�tw sunj ke sunj ke e�nai isodÔname .(1) DmIK;IB! (X) = 0,(2) dmIK;IB! (X) = 0 kai(3) Ind(X) = 0.Apìdeixh. ProkÔptei �mesa apì ti Prot�sei 4.3.6 kai 4.3.7. �

66 Kef�laio 44.4 Jewr mata ajro�smato kai ginomènou4.4.1 Je¸rhma. 'Estw IK kl�sh uposunìlwn tètoia ¸ste (K;X) 2 IK gia k�je q¸roX kai gia k�je kleistì uposÔnolo K tou X. E�n o q¸ro X e�nai ènwsh kleist¸nuposunìlwn Fi, i 2 �, ètsi ¸ste dmIK;IBIE;� (Fi) � � 2 O, tìte dmIK;IBIE;� (X) � �.Apìdeixh. Ef> ìson dmIK;IBIE;� (Fi) � �, apì th sunj kh (3) tou OrismoÔ 4.2.2 prokÔpteiìti Fi = [fQij : j 2 �g ètsi ¸ste gia k�je j 2 � na èqoume:(a) to uposÔnolo Qij tou Fi e�nai kleistì,(b) (Qij; Fi) 2 IK kai(g) DmIK;IBIE;� (Qij) � �.Ef> ìson to uposÔnolo Qij tou X e�nai kleistì sto X, èqoume (Qij; X) 2 IK. Ep�sh ,X = [fQij : i; j 2 �g. Sunep¸ , dmIK;IBIE;� (X) � �. �4.4.2 Orismì . Lègetai ìti:(1) mia kl�sh IK uposunìlwn kai(2) mia kl�sh IB b�sewne�nai kleistè w pro ta ginìmena, e�n èqoume ant�stoiqa:(1) (QX �QY ; X � Y ) 2 IK gia k�je (QX ; X), (QY ; Y ) 2 IK kai(2) (BX�Y ; X � Y ) 2 IB gia k�je (BX ; X), (BY ; Y ) 2 IB, ìpouBX�Y = fUX � UY : UX 2 BX ; UY 2 BY g:4.4.3 Prìtash. Gia k�je dÔo q¸rou X kai Y èqoume:(4:2) DmIK;IB� (X � Y ) � DmIK;IB� (X)(+)DmIK;IB� (Y )kai(4:3) dmIK;IB� (X � Y ) � dmIK;IB� (X)(+)dmIK;IB� (Y );ìpou IK kai IB e�nai kl�sei kleistè w pro ta ginìmena.Apìdeixh. Ja apode�xoume thn prìtash me epagwg . AnDmIK;IB� (X)(+)DmIK;IB� (Y ) = �1  dmIK;IB� (X)(+)dmIK;IB� (Y ) = �1, tìte X = Y = ; kai epomènw DmIK;IB� (X � Y ) = �1  dmIK;IB� (X � Y ) = �1, ant�stoiqa. Upojètoume ìti h anisìthta (4.2) e�nai alhj  giak�je dÔo q¸rou X kai Y me DmIK;IB� (X)(+)DmIK;IB� (Y ) < � kai h anisìthta (4.3) e�naialhj  gia k�je dÔo q¸rou X kai Y me dmIK;IB� (X)(+)dmIK;IB� (Y ) < �, ìpou � e�nai èna stajerì diataktikì arijmì .

Diast�sei tou tÔpou Ind 67'Estw X kai Y dÔo q¸roi me DmIK;IB� (X)(+)DmIK;IB� (Y ) = �. ApodeiknÔoume ìtiDmIK;IB� (X � Y ) � �. E�n DmIK;IB� (X) = �1   DmIK;IB� (Y ) = �1, tìte X � Y = ; kaisunep¸ DmIK;IB� (X � Y ) = �1 < �. 'Estw DmIK;IB� (X) = � kai DmIK;IB� (Y ) = , ìpou �, 2 O. Tìte, up�rqoun IB-b�sei BX � fUÆ : Æ 2 �g kai BY � fVÆ : Æ 2 �g twn X kai Y ,ant�stoiqa, ètsi ¸ste na ikanopoie�tai h sunj kh (2) tou orismoÔ 4.2.2. Ef> ìson h kl�shIB e�nai kleist  w pro ta ginìmena, to sÔnolo BX�Y = fUÆ � VÆ0 : Æ; Æ0 2 �g e�nai miaIB-b�sh tou X � Y . Upojètoume ìti UÆ1 � VÆ01 , UÆ2 � VÆ02 2 BX�Y kaiCl(UÆ1 � VÆ01) \ Cl(UÆ2 � VÆ02) = ;:'EqoumeCl(UÆ1 � VÆ01) \ Cl(UÆ2 � VÆ02) = (Cl(UÆ1)� Cl(VÆ01)) \ (Cl(UÆ2)� Cl(VÆ02))= (Cl(UÆ1) \ Cl(UÆ2))� (Cl(VÆ01) \ Cl(VÆ02))= ;:E�n Cl(UÆ1)\Cl(UÆ2) = ;, tìte up�rqei èna upìqwro L tou X pou diaqwr�zei ta sÔnolaCl(UÆ1) kai Cl(UÆ2) me dmIK;IB� (L) < �. Sunep¸ , up�rqoun dÔo anoikt� uposÔnola WÆ1kai HÆ2 tou X ètsi ¸ste:(a) Cl(UÆ1) � WÆ1 , Cl(UÆ2) � HÆ2 ,(b) WÆ1 \HÆ2 = ; kai(g) X n L = WÆ1 [HÆ2 .'Estw W =WÆ1 � Y , H = HÆ2 � Y kai P = L� Y . Tìte, èqome:Cl(UÆ1 � VÆ01) = Cl(UÆ1)� Cl(VÆ01) � W ,Cl(UÆ2 � VÆ02) = Cl(UÆ2)� Cl(VÆ02) � H,W \H = ; kai (X � Y ) n P = W [H,pou shma�nei ìti to P diaqwr�zei ta sÔnola Cl(UÆ1�VÆ01) kai Cl(UÆ2�VÆ02) sto q¸ro X�Y .Apì thn Prìtash 4.3.1, dmIK;IB� (Y ) � DmIK;IB� (Y ). Sunep¸ ,dmIK;IB� (L)(+)dmIK;IB� (Y ) � dmIK;IB� (L)(+)DmIK;IB� (Y )< �(+) = �:Apì thn upìjesh th epagwg  ,dmIK;IB� (P ) = dmIK;IB� (L� Y ) � dmIK;IB� (L)(+)dmIK;IB� (Y ) < �:

68 Kef�laio 4An�loga me to parap�nw, e�n Cl(VÆ01) \ Cl(VÆ02) = ;, tìte sto q¸ro X � Y mporoÔmena kataskeu�soume èna uposÔnolo P 0 pou diaqwr�zei ta uposÔnola Cl(UÆ1 � VÆ01) kaiCl(UÆ2 � VÆ02) tou X � Y ètsi ¸ste dmIK;IB� (P 0) < �. Sunep¸ ,(4:4) DmIK;IB� (X � Y ) � �:'Estw t¸ra X kai Y dÔo q¸roi me dmIK;IB� (X)(+)dmIK;IB� (Y ) = �. ApodeiknÔoume ìtidmIK;IB� (X � Y ) � �. E�n dmIK;IB� (X) = �1   dmIK;IB� (Y ) = �1, tìte X � Y = ; kaisunep¸ dmIK;IB� (X � Y ) = �1 < �. Upojètoume ìti dmIK;IB� (X) = � kai dmIK;IB� (Y ) = ,ìpou �, 2 O. Apì th sunj kh (3) tou OrismoÔ 4.2.2, èqoume:(a) X = [fQXi : i 2 �g, ìpou to uposÔnolo QXi tou X e�nai kleistì, (QXi ; X) 2 IK kaiDmIK;IB� (QXi ) � � kai(b) Y = [fQYi : i 2 �g, ìpou to uposÔnolo QYi tou Y e�nai kleistì, (QYi ; Y ) 2 IK kaiDmIK;IB� (QYi ) � .ParathroÔme ìti X � Y = [fQXi �QYj : i; j 2 �g;to uposÔnolo QXi �QYj tou X � Y e�nai kleistì kaiDmIK;IB� (QXi )(+)DmIK;IB� (QYj ) � �(+) = �; i; j 2 �:Ef> ìson h kl�sh IK e�nai kleist  w pro ta ginìmena, (QXi �QYj ; X�Y ) 2 IK. Jètonta sth sqèsh (4.4) X = QXi kai Y = QYj , èqoumeDmIK;IB� (QXi �QYj ) � �; i; j 2 �:Sunep¸ , apì th sunj kh (3) tou OrismoÔ 4.2.2,dmIK;IB� (X � Y ) � �:Profan¸ h prìtash e�nai alhj  e�nDmIK;IB� (X)(+)DmIK;IB� (Y ) = dmIK;IB� (X)(+)dmIK;IB� (Y ) =1: �4.5 Idiìthta th kajolikìthta 4.5.1 Sumbolismì . Gia k�je � 2 f�1g [ !, sumbol�zoume meIP(dmIK;IBIE;� � �) kai IP(DmIK;IBIE;� � �)ti kl�sei ìlwn twn q¸rwn X me dmIK;IBIE;� (X) � � kai DmIK;IBIE;� (X) � �, ant�stoiqa.

Diast�sei tou tÔpou Ind 694.5.2 Je¸rhma. 'Estw IB mia koresmènh kl�sh b�sewn, IE mia koresmènh IB-hereditary-separated kl�sh q¸rwn kai IK mia koresmènh kl�sh uposunìlwn me (X;X) 2 IK gia k�jeq¸ro X. Tìte, gia k�je � 2 f�1g [ ! oi kl�sei IP(dmIK;IBIE;� � �) kai IP(DmIK;IBIE;� � �)e�nai koresmène .Apìdeixh. Ja apode�xoume to je¸rhma me epagwg  sto �. 'Estw � = �1. Tìte, èna q¸ro X an kei sthn kl�sh IP(DmIK;IBIE;� � �1) e�n kai mìnon e�n o X an kei sthn kl�shIE, dhlad  IP(DmIK;IBIE;� � �1) = IE:Sunep¸ , h kl�sh IP(DmIK;IBIE;� � �1) e�nai koresmènh. Omo�w , h kl�sh IP(dmIK;IBIE;� � �1)e�nai koresmènh.'Estw � 2 !. Upojètoume ìti oi kl�sei IP(dmIK;IBIE;� � m) kai IP(DmIK;IBIE;� � m),ìpou m 2 f�1g [ �, einai koresmène . ApodeiknÔoume ìti oi kl�sei IP(DmIK;IBIE;� � �) kaiIP(dmIK;IBIE;� � �) e�nai ep�sh koresmène . Pr¸ta apodeiknÔoume ìti h kl�sh IP(DmIK;IBIE;� � �)e�nai koresmènh.'Estw S mia diktuwmènh oikogèneia apì stoiqe�a th IP(DmIK;IBIE;� � �). Gia k�je X 2 Sèstw BX � fV X" : " 2 �g mia diktuwmènh IB-b�sh tou X pou ikanopoie� th sunj kh (2)tou OrismoÔ 4.2.2. Tìte, up�rqoun(i) èna diktuwmèno sÔnolo fLX� : � 2 �g apì uposÔnola tou X,(ii) dÔo diktuwmèna sÔnola fWX� : � 2 �g kai fOX� : � 2 �g apì anoikt� uposÔnola touX kai(iii) mia 1-1 apeikìnish ' tou � � � ep� tou �ètsi ¸ste:(1) Gia k�je "1; "2 2 � kai � = '("1; "2) èqoume(iv) Cl(V X"1 ) � WX� , Cl(V X"2 ) � OX� ,(v) WX� \OX� = ; kai(vi) X n LX� =WX� [OX� ,sthn per�ptwsh, pou e�nai Cl(V X"1 ) \ Cl(V X"2 ) = ; kai LX� = ; sthn per�ptwsh, pou e�naiCl(V X"1 ) \ Cl(V X"2 ) 6= ;.(2) Gia k�je � 2 � , dmIK;IBIE;� (LX� ) � �� 1.Gia k�je � 2 � jètoumeL� = fLX� : X 2 Sg; W� = fWX� : X 2 Sg kai O� = fOX� : X 2 Sg:

70 Kef�laio 4Apì thn idiìthta (2) prokÔptei ìti h L� e�nai mia diktuwmènh oikogèneia apì stoiqe�a th kl�sh IP��1 � IP(dmIK;IBIE;� � �� 1). Apì thn upìjesh th epagwg  , h kl�sh IP��1 e�naikoresmènh. Sunep¸ , up�rqei èna arqikì sun-shm�di M+L� th L� pou antistoiqe� sthnkl�sh IP��1. Sumbol�zoume me M� èna sun-shm�di th S tètoio ¸ste to �qno tou ep� th L� na e�nai mia sun-epèktash tou sun-shmadioÔ M+L� .JewroÔme th sun-diktÔwshN � ffV X" : " 2 �g : X 2 Sgth IB-sun-b�sh B � fBX : X 2 Sg th S. Ef> ìson h kl�sh IB e�nai koresmènh kl�shb�sewn, up�rqei èna arqikì sun-shm�di M+IB th S pou antistoiqe� sthn sun-diktÔwsh Nth B kai sthn kl�sh IB. Eidikìtera, to sun-shm�di M+IB e�nai sun-epèktash tou N.Apì thn Prìtash 3.1.5(2), up�rqei èna sun-shm�di M+ th S, to opo�o e�nai sun-epèktash twn sun-shmadi¸n M+IB kai M� gia k�je � 2 � . Eidikìtera, to sun-shm�di M+e�nai sun-epèktash tou N. Ja apode�xoume ìti to M+ e�nai èna arqikì sun-shm�di th Spou antistoiqe� sthn kl�sh IP(DmIK;IBIE;� � �). Pr�gmati, èstwM � ffUXÆ : Æ 2 �g : X 2 Sgmia auja�reth sunepèktash tou M+. Tìte, to sun-shm�di M e�nai sun-epèktash twn sun-shmadi¸n M+IB, N kai M� gia k�je � 2 � . Subol�zoume me # mia endeiktik  sun�rthsh apìto N sto M. Tìte, gia k�je X 2 S, V X" = UX#("), " 2 � . Profan¸ , to sun-shm�di MjL�e�nai mia sun-epèktash tou sun-shmadioÔM+L� th L�. 'Estw R+IB mia arqik  oikogèneia apìsqèsei isodunam�a ep� th S pou antistoiqe� sto sun-shm�diM, sth sun-diktÔwsh N th B kai sthn kl�sh IB. 'Estw ep�sh R+L� mia arqik  oikogèneia apì sqèsei isodunam�a ep�th L� pou antistoiqe� sto sun-shm�di MjL� kai sth kl�sh IP��1. Sumbol�zoume me R�thn oikogèneia apì sqèsei isodunam�a ep� th S tètoia ¸ste to �qno th ep� th L� nae�nai h oikogèneia R+L� .Apì thn Prìtash 3.1.5(1), up�rqei mia epitrept  oikogèneia R+ apì sqèsei isodunam�-a ep� th S, h opo�a e�nai telik¸ leptìterh apì ti oikogèneie R+IB kai R� gia k�je � 2 � .Eidikìtera, h R+ e�nai M-epitrept . Qwr� periorismì th genikìthta , mporoÔme na upo-jèsoume ìti h R+ e�nai (M;W�)-epitrept , (M;O�)-epitrept , (M;Co(W�))-epitrept kai (M;Co(O�))-epitrept . Ja apode�xoume ìti h R+ e�nai mia arqik  oikogèneia th Spou antistoiqe� sto sun-shm�di M th S kai sthn kl�sh IP(DmIK;IBIE;� � �). Pro toÔtojewroÔme mia auja�reth epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, h opo�ae�nai telik¸ leptìterh apì thn R+, kai apodeiknÔoume ìti gia k�je L 2 C}(R) o q¸ro

Diast�sei tou tÔpou Ind 71T(L) an kei sthn kl�sh IP(DmIK;IBIE;� � �). 'Estw L 2 C}(R). Ef> ìson h kl�sh IB e�-nai koresmènh, èqoume (BL};#(�);T(L)) 2 IB. ApodeiknÔoume ìti h b�sh BL};#(�) tou T(L)ikanopoie� th sunj kh (2) tou OrismoÔ 4.2.2, dhlad  gia k�je UTÆ1(H1) kai UTÆ2(H2) th BL};#(�) (ìpou H1, H2 � L), me(4:5) ClT(L)(UTÆ1(H1)) \ ClT(L)(UTÆ2(H2)) = ;up�rqei èna upìqwro L tou q¸rou T(L) pou diaqwr�zei ta sÔnola ClT(L)(UTÆ1(H1)) kaiClT(L)(UTÆ2(H2)) me dmIK;IBIE;� (L) � �� 1.JewroÔme dÔo stoiqe�a UTÆ1(H1) kai UTÆ2(H2) th BL};#(�) pou ikanopoioÔn th sqèsh(4.5). Pr¸ta upojètoume ìti H1 \H2 = ;. Tìte,(vii) ClT(L)(UTÆ1(H1)) � T(H1), ClT(L)(UTÆ2(H2)) � T(L nH1),(viii) T(H1) \ T(L nH1) = ; kai(ix) T(L) = T(H1) [ T(L nH1).Sunep¸ , to kenì sÔnolo diaqwr�zei ta ClT(L)(UTÆ1(H1)) kai ClT(L)(UTÆ2(H2)). Ef> ìsondmIK;IBIE;� (;) = �1 < �, èqoume DmIK;IBIE;� (T(L)) � �. T¸ra, upojètoume ìti H1 \ H2 6= ;.'Estw H = H1 \ H2, #�1(Æ1) = "1, #�1(Æ2) = "2 kai � = '("1; "2). ApodeiknÔoumeìti to sÔnolo T(HjL�) diaqwr�zei ta sÔnola ClT(L)(UTÆ1(H1)) kai ClT(L)(UTÆ2(H2)), kaidmIK;IBIE;� (T(HjL�)) � �� 1 < �.Ef> ìson h kl�sh IP��1 e�nai koresmènh kl�sh q¸rwn, o upìqwro T(HjL�) touT(MjL� ;RjL�) an kei sthn kl�sh IP��1. Epomènw ,dmIK;IBIE;� (T(HjL�)) � �� 1 < �:ApodeiknÔoume ìti to uposÔnolo T(HjL�) tou T(L) diaqwr�zei ta sÔnola ClT(L)(UTÆ1(H1))kai ClT(L)(UTÆ2(H2)). Upojètoume ìti X 2 H. Epeid  ta uposÔnola Cl(V X"1 ) kai Cl(V X"2 )tou X e�nai xèna metaxÔ tou , apì th sunj kh (1) èqoume(x) Cl(V X"1 ) � WX� , Cl(V X"2 ) � OX� ,(xi) WX� \OX� = ; kai(xii) X n LX� =WX� [OX� .Apì ti parap�nw sqèsei prokÔptei ìti(xiii) ClT(L)(UTÆ1(H)) � T(HjW�) = TjW� \ T(H),ClT(L)(UTÆ2(H)) � T(HjO�) = TjO� \ T(H),(xiv) T(HjW�) \ T(HjO�) = ; kai(xv) T(H) n T(HjL�) = T(HjW�) [ T(HjO�).

72 Kef�laio 4Ef> ìson o periorismì W� th S e�nai anoiktì kai h oikogèneia R e�nai (M;Co(W�))-epitrept , apì thn Prìtash 2.4.32, to uposÔnolo TjW� tou T e�nai anoiktì. Omo�w ,to uposÔnolo TjO� tou T e�nai anoiktì. Ep�sh , epeid  to uposÔnolo T(H) tou T e�naianoiktì kai T(H) � T(L), ta sÔnola T(HjW�) kai T(HjO�) e�nai anoikt� sto T(L).Jètonta W = T(H1 nH) [ T(HjWn)kai O = T(L nH1) [ T(HjOn)èqoume(xvi) ClT(L)(UTÆ1(H1)) � W , ClT(L)(UTÆ2(H2)) � O,(xvii) W \O = ; kai(xviii) T(L) n T(HjL�) = W [O.'Ara, o upìqwro T(HjL�) tou q¸rou T(L) diaqwr�zei ta sÔnola ClT(L)(UTÆ1(H1)) kaiClT(L)(UTÆ1(H2)). Sunep¸ , h kl�sh IP(DmIK;IBIE;� � �) e�nai koresmènh.T¸ra, ja apode�xoume ìti h kl�sh IP(dmIK;IBIE;� � �) e�nai koresmènh. 'Estw S miadiktuwmènh oikogèneia apì stoiqe�a th IP(dmIK;IBIE;� � �). Gia k�je X 2 S up�rqei ènadiktuwmèno sÔnolo fQXi : i 2 �g apì uposÔnola tou X ètsi ¸ste:(3) X = [fQXi : i 2 �g.(4) Gia k�je i 2 �, to uposÔnolo QXi tou X e�nai kleistì kai (QXi ; X) 2 IK.(5) Gia k�je i 2 �, DmIK;IBIE;� (QXi ) � �.Jètoume Qi = fQXi : X 2 Sg; i 2 �:Apì to pr¸to mèro th apìdeixh , h kl�sh IP � IP(DmIK;IBIE;� � �) e�nai koresmènh. Apì thnidiìthta (5), h Qi e�nai mia diktuwmènh oikogèneia apì stoiqe�a th kl�sh IP. Sunep¸ ,up�rqei èna arqikì sun-shm�di M+Qi th Qi pou antistoiqe� sthn kl�sh IP. Sumbol�zoumemeMi èna sun-shm�di th S tètoio ¸ste to �qno tou ep� th Qi na e�nai sun-epèktash tousun-shmadioÔ M+Qi . Apì thn idiìthata (4), o periorismì Qi th S e�nai IK-periorismì .Epeid  h kl�sh IK e�nai koresmènh kl�sh uposunìlwn, gia k�je i 2 � up�rqei èna arqikìsun-shm�di M+IK;i th S pou antistoiqe� ston periorismì Qi kai sthn kl�sh IK.Apì thn Prìtash 3.1.5(2), up�rqei èna sun-shm�di M+ th S, to opo�o e�nai sun-epèktash twn sun-shmadi¸n Mi kai M+IK;i gia k�je i 2 �. ApodeiknÔoume ìti to M+ e�nai

Diast�sei tou tÔpou Ind 73èna arqikì sun-shm�di th S pou antistoiqe� sthn kl�sh IP(dmIK;IBIE;� � �). Pr�gmati, èstwM � ffUXÆ : Æ 2 �g : X 2 Sgmia auja�reth sun-epèktash tou M+. Tìte, to M e�nai sun-epèktash twn sun-shmadi¸nMi kai M+IK;i th S kai to MjQi e�nai sun-epèktash tou sun-shmadioÔ M+Qi th Qi, i 2 �.'Estw R+Qi mia arqik  oikogèneia apì sqèsei isodunam�a ep� th Qi pou antistoiqe� stosun-shm�di MjQi kai sthn kl�sh IP. Sumbol�zoume me Ri thn oikogèneia apì sqèsei isodunam�a ep� th S tètoia ¸ste to �qno th ep� th Qi na e�nai h oikogèneia R+Qi . 'Estwep�sh R+IK;i mia arqik  oikogèneia apì sqèsei isodunam�a ep� th S pou antistoiqe� stosun-shm�di M, ston periorismì Qi kai sthn kl�sh IK.Apì thn Prìtash 3.1.5(1), up�rqei mia epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S, h opo�a e�nai telik¸ leptìterh apì ti oikogèneie Ri kai R+IK;i, i 2 �. 'Ara,h oikogèneia R+ e�nai M-epitrept . Ja apode�xoume ìti h R+ e�nai mia arqik  oikogèneiath S pou antistoiqe� sto sun-shm�di M th S kai sthn kl�sh IP(dmIK;IBIE;� � �). Pro toÔto, jewroÔme mia auja�reth epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, h opo�a e�nai telik¸ leptìterh th R+. Tìte, h R e�nai telik¸ leptìterh apì ti oikogèneie Ri kai R+IK;i gia k�je i 2 �. Arke� na apode�xoume ìti gia k�je L 2 C}(R),T(L) 2 IP(dmIK;IBIE;� � �).'Estw L 2 C}(R). ApodeiknÔoume ìti T(L) = [fTi(L) : i 2 �g ètsi ¸ste:(xix) to uposÔnolo Ti(L) tou T(L) e�nai kleistì,(xx) (Ti(L);T(L)) 2 IK kai(xxi) DmIK;IBIE;� (Ti(L)) � �, i 2 �.Jètoume Ti(L) = T(LjQi); i 2 �:To uposÔnolo T(LjQi) tou T(L) e�nai kleistì kai T(L) = [fT(LjQi) : i 2 �g. Ef> ìsonh kl�sh IK e�nai koresmènh kl�sh uposunìlwn, (T(LjQi);T(L)) 2 IK. Ep�sh , ef> ìsonh kl�sh IP e�nai koresmènh, o upìqwro T(LjQi) tou T(MjQi ;RjQi) an kei sthn kl�shIP. Opìte, DmIK;IBIE;� (T(LjQi)) � �. Sunep¸ , apì th sunj kh (3) tou OrismoÔ 4.2.2,dmIK;IBIE;� (T(L)) � �. 'Ara, h kl�sh IP(dmIK;IBIE;� � �) e�nai koresmènh. �4.5.3 Pìrisma. Gia k�je � 2 ! sti kl�sei IP(dmIK;IBIE;� � �) kai IP(DmIK;IBIE;� � �)up�rqoun kajolik� stoiqe�a.Apìdeixh. ProkÔptei �mesa apì thn Prìtash 3.1.7. �4.5.4 Pìrisma. 'Estw IP mia apì ti parak�tw kl�sei :

74 Kef�laio 4(1) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn me b�ro � � ,(2) h kl�sh ìlwn twn (pl rw ) kanonik¸n ountable-dimensional q¸rwn me b�ro � � ,(3) h kl�sh ìlwn twn (pl rw ) kanonik¸n strongly ountable-dimensional q¸rwn me b�ro � � ,(4) h kl�sh ìlwn twn (pl rw ) kanonik¸n lo ally �nite-dimensional q¸rwn me b�ro � �kai(5) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn X me w(X) � � kai ind(X) � � 2 �+.Tìte, gia k�je � 2 ! sti kl�sei IP(dmIK;IBIE;� � �)\ IP kai IP(DmIK;IBIE;� � �)\ IP up�rqounkajolik� stoiqe�a.Apìdeixh. ProkÔptei �mesa apì thn Prìtash 3.1.6. �

Kef�laio 5Diast�sei -sunart sei jèsew toutÔpou indSto kef�laio autì melet¸ntai diast�sei -sunart sei jèsew tou tÔpou ind. Oi dia-st�sei autè or�sjhkan sto bibl�o [37℄ kai melet jhkan mìno w pro thn idiìthta th kajolikìthta . Sto kef�laio autì d�nontai sqèsei metaxÔ twn diast�sewn-sunart sewnjèsew tou tÔpou ind kai apodeiknÔontai basikè idiìthte th Jewr�a Diast�sewn giati sunart sei autè . Ta apotelèsmata tou kefala�ou autoÔ e�nai ìla prwtìtupa.Se ì,ti akolouje� me th lèxh q¸ro ja ennooÔme ènan T0-q¸ro me b�ro � � .5.1 Basiko� orismo�5.1.1 Orismì . (Blèpe [37℄) 'Estw Q èna uposÔnolo enì q¸rou X. Mia oikogèneia Bapì anoikt� uposÔnola tou X (sumperilambanomènwn twn X kai ;) kale�tai p-b�sh giato Q sto X, e�n to sÔnolo fQ\U : U 2 Bg e�nai mia b�sh tou upoq¸rou Q. Mia p-b�shgia to Q sto X kale�tai pos-b�sh, e�n gia k�je x 2 Q kai gia k�je anoikt  perioq  Utou x sto X up�rqei èna stoiqe�o V th B ètsi ¸ste x 2 V � U . Mia p-b�sh gia to Qsto X kale�tai ps-b�sh, e�n h oikogèneia B e�nai mia b�sh tou q¸rou X.5.1.2 Orismì . (Blèpe [37℄) JewroÔme th sun�rthsh p0-ind me ped�o orismoÔ thn kl�shìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thnkl�sh O [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) p0-ind(Q;X) = �1 e�n kai mìnon e�n Q = X = ;.(2) p0-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia p-b�sh B gia to Q stoX ètsi ¸ste gia k�je U 2 B na èqoumep0-ind(Q \ BdX(U);BdX(U)) < �:75

76 Kef�laio 55.1.3 Parat rhsh. H sunj kh (2) tou OrismoÔ 5.1.2 e�nai isodÔnamh me thn parak�twsunj kh:(2') p0-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je x 2 Q kai gia k�jeanoikt  perioq  V tou x sto X up�rqei èna anoiktì uposÔnolo U tou X ètsi ¸stex 2 Q \ U � Q \ V kai p0-ind(Q \ BdX(U);BdX(U)) < �.5.1.4 Orismì . (Blèpe [37℄) JewroÔme th sun�rthsh p1-ind me ped�o orismoÔ thn kl�shìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thnkl�sh O [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) p1-ind(Q;X) = �1 e�n kai mìnon e�n Q = ;.(2) p1-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia p-b�sh B gia to Q stoX ètsi ¸ste gia k�je U 2 B na èqoumep1-ind(Q \ BdX(U); X) < �:5.1.5 Parat rhsh. H sunj kh (2) tou OrismoÔ 5.1.4 e�nai isodÔnamh me thn parak�twsunj kh:(2') p1-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je x 2 Q kai gia k�jeanoikt  perioq  V tou x sto X up�rqei èna anoiktì uposÔnolo U tou X ètsi ¸stex 2 Q \ U � Q \ V kai p1-ind(Q \ BdX(U); X) < �.5.1.6 Orismì . (Blèpe [37℄) JewroÔme th sun�rthsh pos0-ind me ped�o orismoÔ thn kl�shìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thnkl�sh O [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) pos0-ind(Q;X) = �1 e�n kai mìnon e�n Q = X = ;.(2) pos0-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia pos-b�sh B gia to Qsto X ètsi ¸ste gia k�je U 2 B na èqoumepos0-ind(Q \ BdX(U);BdX(U)) < �:5.1.7 Parat rhsh. H sunj kh (2) tou OrismoÔ 5.1.6 e�nai isodÔnamh me thn parak�twsunj kh:(2') pos0-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je x 2 Q kai gia k�jeanoikt  perioq  V tou x sto X up�rqei èna anoiktì uposÔnolo U tou X ètsi ¸stex 2 U � V kai pos0-ind(Q \ BdX(U);BdX(U)) < �.

Diast�sei -sunart sei jèsew tou tÔpou ind 775.1.8 Orismì . (Blèpe [37℄) JewroÔme th sun�rthsh pos1-ind me ped�o orismoÔ thn kl�shìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thnkl�sh O [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) pos1-ind(Q;X) = �1 e�n kai mìnon e�n Q = ;.(2) pos1-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia pos-b�sh B gia to Qsto X ètsi ¸ste gia k�je U 2 B na èqoumepos1-ind(Q \ BdX(U); X) < �:5.1.9 Parat rhsh. H sunj kh (2) tou OrismoÔ 5.1.8 e�nai isodÔnamh me thn parak�twsunj kh:(2') pos1-ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je x 2 Q kai gia k�jeanoikt  perioq  V tou x sto X up�rqei èna anoiktì uposÔnolo U tou X ètsi ¸stex 2 U � V kai pos1-ind(Q \ BdX(U); X) < �.5.1.10 Parat rhsh. E�n stou orismoÔ 5.1.6 kai 5.1.8 ant� gia thn pos-b�sh B jewr -soume mia ps-b�sh, tìte oi diast�sei -sunart sei posi-ind, i 2 f0; 1g, ja sumbol�zontaime psi-ind. Shmei¸noume ìti h sun�rthsh-di�stash pos1-ind e�nai h uperpeperasmènh epè-ktash th sqetik  mikr  epagwgik  di�stash pou èqei doje� sti [69℄ kai [70℄ (blèpeep�sh [30℄).5.1.11 Parat rhsh. E�nai gnwstì (blèpe [37℄) ìti gia k�je zeÔgo (Q;X) èqoumepi-ind(Q;X) 2 f�1;1g[ �+,posi-ind(Q;X) 2 f�1;1g[ �+ kaipsi-ind(Q;X) 2 f�1;1g[ �+, i = 0; 1,ìpou � e�nai to b�ro tou X kai �+ e�nai o mikrìtero plhj�rijmo pou e�nai megalÔtero apì to � . Ep�sh , èan me df sumbol�soume mia apì ti parap�nw sunart sei -diast�sei ,tìte df(Q;X) =1 e�n kai mìnon e�n h anisìthta df(Q;X) � � den e�nai alhj  gia k�je� 2 �+ [ f�1g.Oi parap�nw diast�sei or�sjhkan sto [37℄ me to ìnoma {positional dimension like-fun tions of the type ind} (diast�sei -sunart sei jèsew tou tÔpou ind). Oi sunar-t sei autè melet jhkan mìno ìson afor� thn idiìthta th kajolikìthta , dhlad  e�ndf e�nai mia apì ti parap�nw sunart sei kai � 2 �+ [ f�1g, tìte sthn kl�sh IP ìlwntwn zeug¸n (QX ; X), ìpou QX e�nai èna uposÔnolo enì q¸rou X me df(QX ; X) � �,up�rqei kajolikì stoiqe�o. ('Ena stoiqe�o (QT ; T ) th IP kale�tai kajolikì sthn IP,

78 Kef�laio 5e�n gia k�je (QX ; X) 2 IP up�rqei mia topologik  emfÔteush iXT : X ! T ètsi ¸steiXT (QX) � QT .) Sqetik� me k�poie �lle diast�sei -sunart sei jèsew me to ìnoma{relative dimensions} blèpe gia par�deigma ti [12℄ kai [13℄.5.2 Sqèsei metaxÔ twn diast�sewn jèsew tou tÔ-pou ind kai �llwn diast�sewn5.2.1 Prìtash. Gia k�je uposÔnolo Q enì q¸rou X èqoumeind(Q) � pi-ind(Q;X); i 2 f0; 1g:Apìdeixh. ApodeiknÔoume ìti(5:1) ind(Q) � p0-ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw p0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.1) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.1) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me p0-ind(QY ; Y ) < �. Ja apode�xoume ìtiind(Q) � �. Ef> ìson p0-ind(Q;X) = �, up�rqei mia p-b�sh B gia to Q sto X ètsi ¸stegia k�je U 2 B na èqoume p0-ind(Q \ BdX(U);BdX(U)) < �:Arke� na apode�xoume ìti ind(BdQ(Q \ U)) < � gia k�je U 2 B. Ef> ìsonBdQ(Q \ U) � Q \ BdX(U);apì thn upìjesh th epagwg  èqoumeind(BdQ(Q \ U)) � ind(Q \ BdX(U))� p0-ind(Q \ BdX(U);BdX(U)) < �:Sunep¸ , ind(Q) � �. �5.2.2 Prìtash. Gia k�je uposÔnolo Q enì q¸rou X èqoumepi-ind(Q;X) � posi-ind(Q;X) � psi-ind(Q;X); i 2 f0; 1g:Apìdeixh. ApodeiknÔoume pr¸ta ìti(5:2) p0-ind(Q;X) � pos0-ind(Q;X):

Diast�sei -sunart sei jèsew tou tÔpou ind 79H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.2) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.2) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me pos0-ind(QY ; Y ) < �. Ja apode�xoume ìtip0-ind(Q;X) � �. Ef> ìson pos0-ind(Q;X) = �, up�rqei mia pos-b�sh B gia to Q stoX ètsi ¸ste gia k�je U 2 B na èqoumepos0-ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,p0-ind(Q \ BdX(U);BdX(U)) � pos0-ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia p-b�sh gia to Q sto X, p0-ind(Q;X) � �.ApodeiknÔoume t¸ra ìti(5:3) pos0-ind(Q;X) � ps0-ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw ps0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.3) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.3) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me ps0-ind(QY ; Y ) < �. Ja apode�xoume ìtipos0-ind(Q;X) � �. Ef> ìson ps0-ind(Q;X) = �, up�rqei mia ps-b�sh B gia to Q sto Xètsi ¸ste gia k�je U 2 B na èqoumeps0-ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,pos0-ind(Q \ BdX(U);BdX(U)) � ps0-ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia pos-b�sh gia to Q sto X, pos0-ind(Q;X) � �. �5.2.3 Prìtash. Gia k�je uposÔnolo Q enì q¸rou X èqoumeps0-ind(Q;X) = ind(X):Apìdeixh. ApodeiknÔoume pr¸ta ìti(5:4) ps0-ind(Q;X) � ind(X):'Estw ind(X) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h sqèsh (5.4) e�naiprofan  . Upojètoume ìti � 2 O kai ìti h sqèsh (5.4) e�nai alhj  gia k�je zeÔgo

80 Kef�laio 5(QY ; Y ) me ind(Y ) < �. Ja apode�xoume ìti ps0-ind(Q;X) � �. Ef> ìson ind(X) = �,up�rqei mia b�sh B tou X ètsi ¸ste gia k�je U 2 B na èqoumeind(BdX(U)) < �:Apì thn upìjesh th epagwg  ,ps0-ind(Q \ BdX(U);BdX(U)) � ind(BdX(U))kai epeid  h B e�nai ep�sh mia ps-b�sh gia to Q sto X, ps0-ind(Q;X) � �.ApodeiknÔoume t¸ra ìti(5:5) ind(X) � ps0-ind(Q;X):'Estw ps0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h anisìthta(5.5) e�nai profan  . Upojètoume ìti � 2 O kai ìti h anisìthta (5.5) e�nai alhj  giak�je zeÔgo (QY ; Y ) me ps0-ind(QY ; Y ) < �. Ja apode�xoume ìti ind(X) � �. Ef> ìsonps0-ind(Q;X) = �, up�rqei mia ps-b�sh B gia to Q sto X ètsi ¸ste gia k�je U 2 B naèqoume ps0-ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,ind(BdX(U)) � ps0-ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia b�sh tou X, ind(X) � �. �5.2.4 Pìrisma. Gia k�je q¸ro X èqoumep0-ind(X;X) = pos0-ind(X;X) = ps0-ind(X;X) = ind(X):Apìdeixh. ProkÔptei �mesa apì ti Prot�sei 5.2.1, 5.2.2 kai 5.2.3. �Ta parak�tw parade�gmata de�qnoun ìti sthn kl�sh ìlwn twn T0-q¸rwn oi anisìthte sti Prot�sei 5.2.1 kai 5.2.2 den mporoÔn na antikatastajoÔn me isìthte .5.2.5 Par�deigma. 'Estw X = fa; bg, Q = fag, K = fbg kai � = f;; fbg; Xg miatopolog�a ep� tou X. Tìte,ind(X) = 1, p0-ind(Q;X) = pos0-ind(Q;X) = 0, p0-ind(K;X) = 0, pos0-ind(K;X) = 1kai ps1-ind(Q;X) = 0.

Diast�sei -sunart sei jèsew tou tÔpou ind 815.2.6 Par�deigma. 'Estw X = fa; b; g, Q = fa; bg kai � = f;; f g; fa; g; fb; g; Xg miatopolog�a ep� tou X. Tìte,ind(Q) = 0; ind(X) = 1, p1-ind(X;X) = pos1-ind(X;X) = 2,pi-ind(Q;X) = posi-ind(Q;X) = 1; i 2 f0; 1g kaips1-ind(K;X) =1.5.2.7 Parat rhsh. Oi sqèsei metaxÔ twn diast�sewn jèsew tou tÔpou ind (blèpe ti parap�nw prot�sei kai parade�gmata) sunoy�zontai sto parak�tw di�gramma, ìpou <<!>>shma�nei <<� >> kai <<9>> shma�nei ìti << genik� � >>.ps0-ind(Q;X) = ind(X)���

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77oooooooooooDi�gramma 5.1H Prìtash 1 tou [69℄ (pou èqei doje� gia peperasmène diast�sei ) mpore� na epektaje�gia uperpeperasmène diast�sei ìpw fa�netai sthn parak�tw prìtash.5.2.8 Prìtash. 'Estw X klhronomik� fusikì q¸ro (dhlad  k�je upìqwro tou Xe�nai fusikì ) kai Q � X. Tìte,ind(Q) = pos1-ind(Q;X):Apìdeixh. Apì ti Prot�sei 5.2.1 kai 5.2.2 arke� na apode�xoume ìti(5:6) pos1-ind(Q;X) � ind(Q):'Estw ind(Q) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h sqèsh (5.6) e�naiprofan  . Upojètoume ìti � 2 O kai ìti h sqèsh (5.6) e�nai alhj  gia k�je zeÔgo

82 Kef�laio 5(QY ; Y ) me ind(QY ) < �. Ja apode�xoume ìti pos1-ind(Q;X) � �. Ef> ìson ind(Q) = �,up�rqei mia pos-b�sh B gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumeind(Q \ BdX(U)) < �(blèpe Problem 2.2.A of [19℄). Apì thn upìjesh th epagwg  ,pos1-ind(Q \ BdX(U);BdX(U)) � ind(Q \ BdX(U)) < �:Sunep¸ , pos1-ind(Q;X) � �. �5.2.9 Pìrisma. 'Estw X klhronomik� fusikì q¸ro (dhlad  k�je upìqwro tou Xe�nai fusikì ) kai Q � X. Tìte,ind(Q) = p1-ind(Q;X) = pos1-ind(Q;X):Apìdeixh. ProkÔptei �mesa apì ti Prot�sei 5.2.1, 5.2.2 kai 5.2.8. �5.2.10 Pìrisma. Gia k�je uposÔnolo Q enì metrikoÔ q¸rou X èqoumeind(Q) = p1-ind(Q;X) = pos1-ind(Q;X):Apìdeixh. ProkÔptei �mesa apì to gegonì ìti k�je metrikì q¸ro e�nai klhronomik�fusikì . �5.3 Jewr mata Upoq¸rou5.3.1 Prìtash. 'Estw i 2 f0; 1g kai Q;K dÔo uposÔnola enì q¸rou X me K � Q.Tìte,(a) pi-ind(K;X) � pi-ind(Q;X),(b) posi-ind(K;X) � posi-ind(Q;X) kai(g) psi-ind(K;X) � psi-ind(Q;X).Apìdeixh. (a) ApodeiknÔoume thn anisìthta(5:7) p0-ind(K;X) � p0-ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw p0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h anisìthta (5.7) profan¸ isqÔei. Upojètoume ìti � 2 O kai ìti h

Diast�sei -sunart sei jèsew tou tÔpou ind 83anisìthta (5.7) e�nai alhj  gia k�je K � Q � X me p0-ind(Q;X) < �. Ja apode�xoumeìti p0-ind(Q;X) � �. Ef> ìson p0-ind(Q;X) = �, up�rqei mia p-b�sh B gia to Q sto Xètsi ¸ste gia k�je U 2 B na èqoumep0-ind(Q \ BdX(U);BdX(U)) < �:Ef> ìson K \ BdX(U) � Q \ BdX(U);apì thn upìjesh th epagwg  ,p0-ind(K \ BdX(U);BdX(U)) � p0-ind(Q \ BdX(U);BdX(U)) < �kai epeid  h B e�nai ep�sh mia p-b�sh gia to K sto X, p0-ind(K;X) � �.(b) ApodeiknÔoume thn anisìthta(5:8) pos0-ind(K;X) � pos0-ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.8) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.8) e�nai alhj  gia k�je K � Q � X me pos0-ind(Q;X) < �. Ja apode�xoume ìtipos0-ind(Q;X) � �. Ef> ìson pos0-ind(Q;X) = �, up�rqei mia p-b�sh B gia to Q stoX ètsi ¸ste gia k�je U 2 B na èqoumepos0-ind(Q \ BdX(U);BdX(U)) < �:Ef> ìson K \ BdX(U) � Q \ BdX(U);apì thn upìjesh th epagwg  ,pos0-ind(K \ BdX(U);BdX(U)) � pos0-ind(Q \ BdX(U);BdX(U)) < �kai epeid  h B e�nai ep�sh mia pos-b�sh gia to K sto X, pos0-ind(K;X) � �.(g) ApodeiknÔoume thn anisìthta(5:9) ps0-ind(K;X) � ps0-ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw ps0-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.9) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.9) e�nai alhj  gia k�je K � Q � X me ps0-ind(Q;X) < �. Ja apode�xoume ìti

84 Kef�laio 5ps0-ind(Q;X) � �. Ef> ìson ps0-ind(Q;X) = �, up�rqei mia ps-b�sh B gia to Q sto Xètsi ¸ste gia k�je U 2 B na èqoumeps0-ind(Q \ BdX(U);BdX(U)) < �:Ef> ìson K \ BdX(U) � Q \ BdX(U);apì thn upìjesh th epagwg  ,ps0-ind(K \ BdX(U);BdX(U)) � ps0-ind(Q \ BdX(U);BdX(U)) < �kai epeid  h B e�nai ep�sh mia ps-b�sh gia to K sto X, ps0-ind(K;X) � �. �5.3.2 Prìtash. 'Estw i 2 f0; 1g, Y èna upìqwro enì q¸rou X kai Q � Y . Tìte,(a) pi-ind(Q; Y ) � pi-ind(Q;X),(b) posi-ind(Q; Y ) � posi-ind(Q;X) kai(g) psi-ind(Q; Y ) � psi-ind(Q;X).Apìdeixh. (a) ApodeiknÔoume thn anisìthta(5:10) p1-ind(Q; Y ) � p1-ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw p1-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.10) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.10) e�nai alhj  gia k�je Q � Y � X me p1-ind(Q;X) < �. Ja apode�xoume ìtip1-ind(Q; Y ) � �. Ef> ìson p1-ind(Q;X) = �, up�rqei mia p-b�sh B gia to Q sto X ètsi¸ste gia k�je U 2 B na èqoumep1-ind(Q \ BdX(U); X) < �:Ef> ìson BdY (U \ Y ) � Y \ BdX(U) � BdX(U);apì thn Prìtash 5.3.1,p1-ind(Q \ BdY (U \ Y ); X) � p1-ind(Q \ BdX(U); X) < �:Ep�sh , apì thn upìjesh th epagwg  ,p1-ind(Q \ BdY (U \ Y ); Y ) � p1-ind(Q \ BdY (U \ Y ); X) < �

Diast�sei -sunart sei jèsew tou tÔpou ind 85kai epeid  to sÔnolo fU \ Y : U 2 Bg e�nai mia p-b�sh gia to Q sto Y , èqoumep1-ind(Q; Y ) � �.(b) ApodeiknÔoume thn anisìthta(5:11) pos1-ind(Q; Y ) � pos1-ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw pos1-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.11) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.11) e�nai alhj  gia k�je Q � Y � X me pos1-ind(Q;X) < �. Ja apode�xoume ìtipos1-ind(Q; Y ) � �. Ef> ìson pos1-ind(Q;X) = �, up�rqei mia pos-b�sh B gia to Q stoX ètsi ¸ste gia k�je U 2 B na èqoumepos1-ind(Q \ BdX(U); X) < �:Ef> ìson BdY (U \ Y ) � Y \ BdX(U) � BdX(U);apì thn Prìtash 5.3.1,pos1-ind(Q \ BdY (U \ Y ); X) � pos1-ind(Q \ BdX(U); X) < �:Ep�sh , apì thn upìjesh th epagwg  ,pos1-ind(Q \ BdY (U \ Y ); Y ) � pos1-ind(Q \ BdY (U \ Y ); X) < �kai epeid  to sÔnolo fU \ Y : U 2 Bg e�nai mia pos-b�sh gia to Q sto Y , èqoumepos1-ind(Q; Y ) � �.(g) ApodeiknÔoume thn anisìthta(5:12) ps1-ind(Q; Y ) � ps1-ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw ps1-ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (5.12) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(5.12) e�nai alhj  gia k�je Q � Y � X me ps1-ind(Q;X) < �. Ja apode�xoume ìtips1-ind(Q; Y ) � �. Ef> ìson ps1-ind(Q;X) = �, up�rqei mia ps-b�sh B gia to Q sto Xètsi ¸ste gia k�je U 2 B na èqoumeps1-ind(Q \ BdX(U); X) < �:Ef> ìson BdY (U \ Y ) � Y \ BdX(U) � BdX(U);

86 Kef�laio 5apì thn Prìtash 5.3.1,ps1-ind(Q \ BdY (U \ Y ); X) � ps1-ind(Q \ BdX(U); X) < �:Ep�sh , apì thn upìjesh th epagwg  ,ps1-ind(Q \ BdY (U \ Y ); Y ) � ps1-ind(Q \ BdY (U \ Y ); X) < �kai epeid  to sÔnolo fU \ Y : U 2 Bg e�nai mia ps-b�sh gia to Q sto Y , èqoumeps1-ind(Q; Y ) � �. �5.3.3 Prìtash. 'Estw Y èna puknì uposÔnolo enì q¸rou X kai Q � Y . Tìte,(a) p1-ind(Q; Y ) = p1-ind(Q;X).(b) pos1-ind(Q; Y ) = pos1-ind(Q;X).Apìdeixh. (a) Apì thn Prìtash 5.3.2 arke� na apode�xoume ìti(5:13) p1-ind(Q;X) � p1-ind(Q; Y ):'Estw p1-ind(Q; Y ) = � 2 O [ f�1;1g. E�n � = �1   � =1, tìte h sqèsh (5.13) e�naiprofan  . Upojètoume ìti � 2 O kai ìti h sqèsh (5.13) e�nai alhj  gia k�je Q � Y � Xme p1-ind(Q; Y ) < �, ìpou to Y e�nai puknì sto X. Ja apode�xoume ìti p1-ind(Q;X) � �.Ef> ìson p1-ind(Q; Y ) = �, up�rqei mia p-b�sh B gia to Q sto Y ètsi ¸ste gia k�jeU 2 B na èqoume p1-ind(Q \ BdY (U); Y ) < �.To sÔnolo ìlwn twn anoikt¸n uposunìlwn V tou X ètsi ¸ste V \ Y 2 B e�nai miap-b�sh gia to Q sto X. Ef> ìson to Y e�nai puknì sto X, gia k�je anoiktì uposunìloV tou X èqoume BdY (V \ Y ) = Y \ BdX(V )kai, epomènw , Q \ BdY (V \ Y ) = Q \ BdX(V ):Apì thn upìjesh th epagwg  , e�n V \ Y 2 B, tìte èqoumep1-ind(Q \ BdX(V ); X) = p1-ind(Q \ BdY (V \ Y ); X)� p1-ind(Q \ BdY (V \ Y ); Y ) < �:Sunep¸ , p1-ind(Q;X) � �.(b) Apì thn Prìtash 5.3.2 arke� na apode�xoume ìti(5:14) pos1-ind(Q;X) � pos1-ind(Q; Y ):

Diast�sei -sunart sei jèsew tou tÔpou ind 87'Estw pos1-ind(Q; Y ) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h sqèsh (5.14)e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (5.14) e�nai alhj  gia k�jeQ � Y � X me pos1-ind(Q; Y ) < �, ìpou to Y e�nai puknì sto X. Ja apode�xoume ìtipos1-ind(Q;X) � �. Ef> ìson pos1-ind(Q; Y ) = �, up�rqei mia pos-b�sh B gia to Q stoY ètsi ¸ste gia k�je U 2 B na èqoume pos1-ind(Q \ BdY (U); Y ) < �.To sÔnolo ìlwn twn anoikt¸n uposunìlwn V tou X ètsi ¸ste V \ Y 2 B e�nai miapos-b�sh gia to Q sto X. Ef> ìson to Y e�nai puknì sto X, gia k�je anoiktì uposunìloV tou X èqoume BdY (V \ Y ) = Y \ BdX(V )kai, epomènw , Q \ BdY (V \ Y ) = Q \ BdX(V ):Apì thn upìjesh th epagwg  , e�n V \ Y 2 B, tìte èqoumepos1-ind(Q \ BdX(V ); X) = pos1-ind(Q \ BdY (V \ Y ); X)� pos1-ind(Q \ BdY (V \ Y ); Y ) < �:Sunep¸ , pos1-ind(Q;X) � �. �5.4 Jewr mata Ajro�smato 5.4.1 Prìtash. 'Estw Q1 kai Q2 dÔo uposÔnola enì q¸rou X. Tìte,(5:15) pos0-ind(Q1 [Q2; X) � pos0-ind(Q1; X)(+)pos0-ind(Q2; X):Apìdeixh. Ja apode�xoume th sqèsh (5.15) me epagwg  ep� tou �, ìpou� = pos0-ind(Q1; X)(+)pos0-ind(Q2; X):E�n � = �1, tìte pos0-ind(Q1; X) = pos0-ind(Q2; X) = �1 pou shma�nei ìtiQ1 [Q2 = X = ;kai epomènw h sqèsh (5.15) isqÔei. Upojètoume ìti gia k�je q¸ro X kai k�je dÔouposÔnola Q1; Q2 tou X h sqèsh (5.15) e�nai alhj  e�npos0-ind(Q1; X)(+)pos0-ind(Q2; X) < �;

88 Kef�laio 5ìpou � e�nai èna stajerì diataktikì arijmì . Ja apode�xoume th sqèsh (5.15) giapos0-ind(Q1; X)(+)pos0-ind(Q2; X) = �:'Estw pos0-ind(Q1; X) = �1kai pos0-ind(Q2; X) = �2;ìpou �1; �2 2 O [ f�1g. E�n èna apì ta stoiqe�a �1; �2 e�nai �so me to �1, tìte kai to�llo e�nai �so me to �1 kai epomènw to � = �1 den e�nai diataktikì arijmì . Sunep¸ ,mporoÔme na upojèsoume ìti �1; �2 2 O.Up�rqei mia pos-b�sh B1 gia to Q1 sto X kai mia pos-b�sh B2 gia to Q2 sto X ètsi¸ste pos0-ind(Q1 \ BdX(U1);BdX(U1)) < �1kai pos0-ind(Q2 \ BdX(U2);BdX(U2)) < �2gia k�je U1 2 B1 kai U2 2 B2. To sÔnolo B � B1 [B2 mia pos-b�sh gia to Q1 [Q2 stoX. 'Estw U 2 B, gia par�deigma, U 2 B1. Tìte,pos0-ind(Q1 \ BdX(U);BdX(U)) < �1kai, apì ti Prot�sei 5.3.1(b) kai 5.3.2(b),pos0-ind(Q2 \ BdX(U);BdX(U)) � pos0-ind(Q2; X) = �2:Apì thn upìjesh th epagwg  , èqoumepos0-ind((Q1 [Q2) \ BdX(U);BdX(U)) =pos0-ind((Q1 \ BdX(U)) [ (Q2 \ BdX(U));BdX(U)) �pos0-ind(Q1 \ BdX(U);BdX(U))(+)pos0-ind(Q2 \ BdX(U);BdX(U)) <�1(+)�2 = �:Sunep¸ , pos0-ind(Q1 [Q2; X) � �. �5.4.2 Prìtash. 'Estw Q1 kai Q2 dÔo uposÔnola enì q¸rou X. Tìte,(5:16) pos1-ind(Q1 [Q2; X) � pos1-ind(Q1; X)(+)pos1-ind(Q2; X) + 1

Diast�sei -sunart sei jèsew tou tÔpou ind 89kai(5:17) ps1-ind(Q1 [Q2; X) � ps1-ind(Q1; X)(+)ps1-ind(Q2; X) + 1:Apìdeixh. Ja apode�xoume th sqèsh (5.16) me epagwg  ep� tou �, ìpou� = pos1-ind(Q1; X)(+)pos1-ind(Q2; X):E�n � = �1, tìte pos1-ind(Q1; X) = pos1-ind(Q2; X) = �1 pou shma�nei ìtiQ1 [Q2 = ;kai epomènw h sqèsh (5.16) isqÔei. Upojètoume ìti gia k�je q¸ro X kai k�je dÔouposÔnola Q1; Q2 tou X h sqèsh (5.16) e�nai alhj  e�npos1-ind(Q1; X)(+)pos1-ind(Q2; X) < �;ìpou � e�nai èna stajerì diataktikì arijmì . Ja apode�xoume th sqèsh (5.16) giapos1-ind(Q1; X)(+)pos1-ind(Q2; X) = �:'Estw pos1-ind(Q1; X) = �1kai pos1-ind(Q2; X) = �2;ìpou �1; �2 2 O [ f�1g. E�n �1 = �1   �2 = �1, tìte Q1 = ;   Q2 = ;, ant�stoiqa kaiepomènw h sqèsh (5.16) isqÔei. Upojètoume ìti �1; �2 2 O.Up�rqei mia pos-b�sh B1 gia to Q1 sto X kai mia pos-b�sh B2 gia to Q2 sto X ètsi¸ste pos1-ind(Q1 \ BdX(U1); X) < �1kai pos1-ind(Q2 \ BdX(U2); X) < �2gia k�je U1 2 B1 kai U2 2 B2. To sÔnolo B � B1 [B2 mia pos-b�sh gia to Q1 [Q2 stoX. 'Estw U 2 B, gia par�deigma, U 2 B1. Tìte,pos1-ind(Q1 \ BdX(U); X) < �1kai, apì thn Prìtash 5.3.1(b),pos1-ind(Q2 \ BdX(U); X) � pos1-ind(Q2; X) = �2:

90 Kef�laio 5Apì thn upìjesh th epagwg  , èqoumepos1-ind((Q1 [Q2) \ BdX(U); X) =pos1-ind((Q1 \ BdX(U)) [ (Q2 \ BdX(U)); X) �pos1-ind(Q1 \ BdX(U); X)(+)pos1-ind(Q2 \ BdX(U); X) + 1 <�1(+)�2 + 1 = � + 1:Sunep¸ , pos1-ind(Q1 [Q2; X) � � + 1.Ja apode�xoume th sqèsh (5.17) me epagwg  ep� tou �, ìpou� = ps1-ind(Q1; X)(+)ps1-ind(Q2; X):E�n � = �1, tìte ps1-ind(Q1; X) = ps1-ind(Q2; X) = �1 pou shma�nei ìtiQ1 [Q2 = ;kai epomènw h sqèsh (5.17) isqÔei. Upojètoume ìti gia k�je q¸ro X kai k�je dÔouposÔnola Q1; Q2 tou X h sqèsh (5.17) e�nai alhj  e�nps1-ind(Q1; X)(+)ps1-ind(Q2; X) < �;ìpou � e�nai èna stajerì diataktikì arijmì . Ja apode�xoume th sqèsh (5.17) giaps1-ind(Q1; X)(+)ps1-ind(Q2; X) = �:'Estw ps1-ind(Q1; X) = �1kai ps1-ind(Q2; X) = �2;ìpou �1; �2 2 O [ f�1g. E�n �1 = �1   �2 = �1, tìte Q1 = ;   Q2 = ;, ant�stoiqa kaiepomènw h sqèsh (5.17) isqÔei. Upojètoume ìti �1; �2 2 O.Up�rqei mia ps-b�sh B1 gia to Q1 sto X kai mia ps-b�sh B2 gia to Q2 sto X ètsi¸ste ps1-ind(Q1 \ BdX(U1); X) < �1kai ps1-ind(Q2 \ BdX(U2); X) < �2

Diast�sei -sunart sei jèsew tou tÔpou ind 91gia k�je U1 2 B1 kai U2 2 B2. To sÔnolo B � B1 [ B2 mia ps-b�sh gia to Q1 [ Q2 stoX. 'Estw U 2 B, gia par�deigma, U 2 B1. Tìte,ps1-ind(Q1 \ BdX(U); X) < �1kai, apì thn Prìtash 5.3.1(g),ps1-ind(Q2 \ BdX(U); X) � ps1-ind(Q2; X) = �2:Apì thn upìjesh th epagwg  , èqoumeps1-ind((Q1 [Q2) \ BdX(U); X) =ps1-ind((Q1 \ BdX(U)) [ (Q2 \ BdX(U)); X) �ps1-ind(Q1 \ BdX(U); X)(+)ps1-ind(Q2 \ BdX(U); X) + 1 <�1(+)�2 + 1 = � + 1:Sunep¸ , ps1-ind(Q1 [Q2; X) � � + 1. �5.5 Apeikon�sei 5.5.1 Prìtash. 'Estw f : X ! Y suneq  apeikìnish kai Q � X. E�n o periorismì f jQ th apeikìnish f ep� tou Q e�nai omoiomorfismì , tìte(5:18) p1-ind(Q;X) � p1-ind(f(Q); Y )kai(5:19) pos1-ind(Q;X) � pos1-ind(f(Q); Y ):Apìdeixh. Ja apode�xoume th sqèsh (5.18) me epagwg  ep� tou stoiqe�oup1-ind(f(Q); Y ) 2 O [ f�1;1g:H sqèsh (5.18) isqÔei e�n p1-ind(f(Q); Y ) = �1   p1-ind(f(Q); Y ) = 1. Upojètoumeìti h sqèsh (5.18) isqÔei e�n p1-ind(f(Q); Y ) < � 2 O kai apodeiknÔoume ìti isqÔei giap1-ind(f(Q); Y ) = �. Up�rqei mia p-b�sh B gia to f(Q) sto Y ètsi ¸ste gia k�jeW 2 Bna èqoume p1-ind(f(Q) \ BdY (W ); Y ) < �:

92 Kef�laio 5To sÔnolo ff�1(W ) : W 2 Bg e�nai mia p-b�sh gia to Q sto X. Arke� na apode�xoumeìti p1-ind(Q \ BdX(f�1(W )); X) < � gia k�je W 2 B. Ef> ìson h apeikìnish f e�naisuneq  , èqoume f(BdX(f�1(W ))) � BdY (W ). Apì thn Prìtash 5.3.1(a),p1-ind(f(Q \ BdX(f�1(W ))); Y ) = p1-ind(f(Q) \ f(BdX(f�1(W ))); Y )� p1-ind(f(Q) \ BdY (W ); Y ) < �:Sunep¸ , apì thn upìjesh th epagwg  ,p1-ind(Q \ BdX(f�1(W )); X) � p1-ind(f(Q) \ BdY (W ); Y ) < �kai epomènw p1-ind(Q;X) � �.Ja apode�xoume th sqèsh (5.19) me epagwg  ep� tou stoiqe�oupos1-ind(f(Q); Y ) 2 O [ f�1;1g:H sqèsh (5.19) isqÔei e�n pos1-ind(f(Q); Y ) = �1   pos1-ind(f(Q); Y ) =1. Upojètoumeìti h sqèsh (5.19) isqÔei e�n pos1-ind(f(Q); Y ) < � 2 O kai apodeiknÔoume ìti isqÔei giapos1-ind(f(Q); Y ) = �. Up�rqei mia pos-b�sh B gia to f(Q) sto Y ètsi ¸ste gia k�jeW 2 B na èqoume pos1-ind(f(Q) \ BdY (W ); Y ) < �:To sÔnolo ff�1(W ) : W 2 Bg e�nai mia pos-b�sh gia to Q sto X. Arke� na apode�xoumeìti pos1-ind(Q \ BdX(f�1(W )); X) < � gia k�je W 2 B. Ef> ìson h apeikìnish f e�naisuneq  , èqoume f(BdX(f�1(W ))) � BdY (W ). Apì thn Prìtash 5.3.1(b),pos1-ind(f(Q \ BdX(f�1(W ))); Y ) = pos1-ind(f(Q) \ f(BdX(f�1(W ))); Y )� pos1-ind(f(Q) \ BdY (W ); Y ) < �:Sunep¸ , apì thn upìjesh th epagwg  ,pos1-ind(Q \ BdX(f�1(W )); X) � pos1-ind(f(Q) \ BdY (W ); Y ) < �kai epomènw pos1-ind(Q;X) � �. �

Kef�laio 6Diast�sei -sunart sei jèsew toutÔpou IndSthn ergas�a [40℄ (blèpe ep�sh [30℄) or�sjhke mia di�stash-sun�rthsh jèsew toutÔpou Ind. Sto kef�laio autì d�nontai kai melet¸ntai kainoÔrgie diast�sei -sunart sei jèsew tou tÔpou Ind kai apodeiknÔontai basikè idiìthte th Jewr�a Diast�sewn giati sunart sei autè . Ta apotelèsmata tou kefala�ou autoÔ e�nai ìla prwtìtupa.Se ì,ti akolouje� me th lèxh q¸ro ja ennooÔme ènan T0-q¸ro me b�ro � � .6.1 Basiko� orismo�6.1.1 Orismì . (Blèpe, gia par�deigma, [45℄) Mia oikogèneia B apì anoikt� uposÔnolaenì q¸rou X kale�tai meg�lh b�sh (big base) tou X, e�n gia k�je zeÔgo (F; U) apìuposÔnola tou X, ìpou to F e�nai kleistì, to U e�nai anoiktì kai F � U up�rqei V 2 Bme thn idiìthta F � V � U .6.1.2 Orismì . 'Estw Q èna uposÔnolo enì q¸rou X. Mia oikogèneia B apì anoikt�uposÔnola tou X kale�tai p(0)-meg�lh b�sh tou Q sto X (p(0)-big base for Q in X),e�n gia k�je zeÔgo (F; U) apì uposÔnola tou X, ìpou to F e�nai kleistì uposÔnolo touX, to U e�nai anoiktì uposÔnolo tou X kai F � Q \ U up�rqei V 2 B me thn idiìthtaF � Q\V � Q\U . Mia p(0)-meg�lh b�sh B tou Q stoX kale�tai pos(0)-meg�lh b�shtou Q sto X (pos(0)-big base for Q in X), e�n gia k�je zeÔgo (F; U) apì uposÔnolatou X, ìpou to F e�nai kleistì uposÔnolo tou X, to U e�nai anoiktì uposÔnolo tou Xkai F � Q \ U up�rqei V 2 B me thn idiìthta F � V � U .6.1.3 Orismì . JewroÔme th sun�rthsh p0(0)-Ind me ped�o orismoÔ thn kl�sh ìlwn twnzeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thn kl�shO [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :93

94 Kef�laio 6(1) p0(0)-Ind(Q;X) = �1 e�n kai mìnon e�n Q = X = ;.(2) p0(0)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia p(0)-meg�lh b�sh Bgia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep0(0)-Ind(Q \ BdX(U);BdX(U)) < �:6.1.4 Parat rhsh. H sunj kh (2) tou OrismoÔ 6.1.3 e�nai isodÔnamh me thn parak�twsunj kh:(2') p0(0)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je zeÔgo (F; U) apìuposÔnola touX, ìpou to F e�nai kleistì uposÔnolo tou X, to U e�nai anoiktì uposÔnolotou X kai F � Q\U up�rqei anoiktì uposÔnolo V tou X ètsi ¸ste F � Q\V � Q\Ukai p0(0)-Ind(Q \ BdX(V );BdX(V )) < �:6.1.5 Orismì . JewroÔme th sun�rthsh p1(0)-Ind me ped�o orismoÔ thn kl�sh ìlwn twnzeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thn kl�shO [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) p1(0)-Ind(Q;X) = �1 e�n kai mìnon e�n Q = ;.(2) p1(0)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia p(0)-meg�lh b�sh Bgia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep1(0)-Ind(Q \ BdX(U); X) < �:6.1.6 Parat rhsh. H sunj kh (2) tou OrismoÔ 6.1.5 e�nai isodÔnamh me thn parak�twsunj kh:(2') p1(0)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je zeÔgo (F; U) apìuposÔnola touX, ìpou to F e�nai kleistì uposÔnolo tou X, to U e�nai anoiktì uposÔnolotou X kai F � Q\U up�rqei anoiktì uposÔnolo V tou X ètsi ¸ste F � Q\V � Q\Ukai p1(0)-Ind(Q \ BdX(V ); X) < �:6.1.7 Parat rhsh. (1) E�n stou orismoÔ 6.1.3 kai 6.1.5 ant� gia thn p(0)-meg�lhb�sh B jewr soume mia pos(0)-meg�lh b�sh B, tìte oi diast�sei -sunart sei pi(0)-Ind,i 2 f0; 1g, ja sumbol�zontai me posi(0)-Ind.(2) Gia k�je q¸ro X èqoumep0(0)-Ind(X;X) = pos0(0)-Ind(X;X) = Ind(X):

Diast�sei -sunart sei jèsew tou tÔpou Ind 956.1.8 Orismì . 'Estw Q èna uposÔnolo enì q¸rou X. Mia oikogèneia B apì anoikt�uposÔnola tou X kale�tai p(1)-meg�lh b�sh tou Q sto X (p(1)-big base for Q in X),e�n to sÔnolo fQ \ U : U 2 Bg e�nai meg�lh b�sh gia ton upìqwro Q. Mia p(1)-meg�lhb�sh B tou Q sto X kale�tai pos(1)-meg�lh b�sh tou Q sto X (pos(1)-big base forQ in X), e�n gia k�je zeÔgo (FQ; U) apì uposÔnola tou X, ìpou to FQ e�nai kleistìuposÔnolo tou Q, to U e�nai anoiktì uposÔnolo tou X kai FQ � U up�rqei V 2 B me thnidiìthta FQ � V � U .6.1.9 Orismì . JewroÔme th sun�rthsh p0(1)-Ind me ped�o orismoÔ thn kl�sh ìlwn twnzeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thn kl�shO [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) p0(1)-Ind(Q;X) = �1 e�n kai mìnon e�n Q = X = ;.(2) p0(1)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia p(1)-meg�lh b�sh Bgia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep0(1)-Ind(Q \ BdX(U);BdX(U)) < �:6.1.10 Parat rhsh. H sunj kh (2) tou OrismoÔ 6.1.9 e�nai isodÔnamh me thn parak�twsunj kh:(2') p0(1)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je zeÔgo (FQ; U)apì uposÔnola tou X, ìpou to FQ e�nai kleistì uposÔnolo tou Q, to U e�nai anoiktìuposÔnolo tou X kai FQ � Q \ U up�rqei èna anoiktì uposÔnolo V tou X ètsi ¸steFQ � Q \ V � Q \ U kaip0(1)-Ind(Q \ BdX(V );BdX(V )) < �:6.1.11 Orismì . JewroÔme th sun�rthsh p1(1)-Ind me ped�o orismoÔ thn kl�sh ìlwntwn zeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n thn kl�shO [ f�1;1g pou ikanopoie� ti parak�tw sunj ke :(1) p1(1)-Ind(Q;X) = �1 e�n kai mìnon e�n Q = ;.(2) p1(1)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n up�rqei mia p(1)-meg�lh b�sh Bgia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep1(1)-Ind(Q \ BdX(U); X) < �:6.1.12 Parat rhsh. H sunj kh (2) tou OrismoÔ 6.1.11 e�nai isodÔnamh me thn parak�twsunj kh:

96 Kef�laio 6(2') p1(1)-Ind(Q;X) � �, ìpou � 2 O, e�n kai mìnon e�n gia k�je zeÔgo (FQ; U)apì uposÔnola tou X, ìpou to FQ e�nai kleistì uposÔnolo tou Q, to U e�nai anoiktìuposÔnolo tou X kai FQ � Q \ U up�rqei èna anoiktì uposÔnolo V tou X ètsi ¸steFQ � Q \ V � Q \ U kai p1(1)-Ind(Q \ BdX(V ); X) < �:6.1.13 Parat rhsh. (1) E�n stou orismoÔ 6.1.9 kai 6.1.11 ant� gia thn p(1)-meg�lhb�sh B jewr soume mia pos(1)-meg�lh b�sh B, tìte oi diast�sei -sunart sei pi(1)-Ind,i 2 f0; 1g, ja sumbol�zontai me posi(1)-Ind. Shmei¸noume ìti h sun�rthsh-di�stashpos1(1)-ind e�nai h uperpeperasmènh epèktash th sqetik  meg�lh epagwgik  di�stash pou èqei doje� sthn ergas�a [40℄ (blèpe ep�sh [30℄).(2) Gia k�je q¸ro X èqoumep0(1)-Ind(X;X) = pos0(1)-Ind(X;X) = Ind(X):6.2 Sqèsei metaxÔ twn diast�sewn jèsew tou tÔ-pou Ind kai �llwn diast�sewn6.2.1 Prìtash. 'Estw i 2 f0; 1g. Gia k�je uposÔnolo Q enì q¸rou X èqoume(1) pi(0)-Ind(Q;X) � pi(1)-Ind(Q;X) kai(2) posi(0)-Ind(Q;X) � posi(1)-Ind(Q;X).E�n epiplèon to uposÔnolo Q tou X e�nai kleistì, tìtepi(0)-Ind(Q;X) = pi(1)-Ind(Q;X)kai posi(0)-Ind(Q;X) = posi(1)-Ind(Q;X):Apìdeixh. (1) ApodeiknÔoume ìti(6:1) p0(0)-Ind(Q;X) � p0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw p0(1)-Ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (6.1) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(6.1) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me p0(1)-Ind(QY ; Y ) < �. Ja apode�xoume ìti

Diast�sei -sunart sei jèsew tou tÔpou Ind 97p0(0)-Ind(Q;X) � �. Ef> ìson p0(1)-Ind(Q;X) = �, up�rqei mia p(1)-meg�lh b�sh B giato Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,p0(0)-Ind(Q \ BdX(U);BdX(U)) � p0(1)-Ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia p(0)-meg�lh b�sh gia to Q sto X, p0(0)-Ind(Q;X) � �.'Estw t¸ra ìti to uposÔnolo Q tou X e�nai kleistì. ApodeiknÔoume ìtip0(0)-Ind(Q;X) = p0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. Arke� na apode�xoume ìti(6:2) p0(1)-Ind(Q;X) � p0(0)-Ind(Q;X):'Estw p0(0)-Ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h sqèsh (6.2)e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.2) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me p0(0)-Ind(QY ; Y ) < �. Ja apode�xoume ìti p0(1)-Ind(Q;X) � �. Ef> ìsonp0(0)-Ind(Q;X) = �, up�rqei mia p(0)-meg�lh b�sh B gia to Q sto X ètsi ¸ste gia k�jeU 2 B na èqoume p0(0)-Ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,p0(1)-Ind(Q \ BdX(U);BdX(U)) � p0(0)-Ind(Q \ BdX(U);BdX(U)):Epeid  to uposÔnolo Q tou X e�nai kleistì, h B e�nai ep�sh mia p(1)-meg�lh b�sh gia toQ sto X. Sunep¸ , p0(0)-Ind(Q;X) � �.(2) ApodeiknÔoume ìti(6:3) pos0(0)-Ind(Q;X) � pos0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0(1)-Ind(Q;X) = � 2 O[f�1;1g. E�n � = �1  � = 1, tìte h sqèsh (6.3) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh(6.3) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me pos0(1)-Ind(QY ; Y ) < �. Ja apode�xoumeìti pos0(0)-Ind(Q;X) � �. Ef> ìson pos0(1)-Ind(Q;X) = �, up�rqei mia pos(1)-meg�lhb�sh B gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumepos0(1)-Ind(Q \ BdX(U);BdX(U)) < �:

98 Kef�laio 6Apì thn upìjesh th epagwg  ,pos0(0)-Ind(Q \ BdX(U);BdX(U)) � pos0(1)-Ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia pos(0)-meg�lh b�sh gia to Q sto X, pos0(0)-Ind(Q;X) � �.'Estw t¸ra ìti to uposÔnolo Q tou X e�nai kleistì. ApodeiknÔoume ìtipos0(0)-Ind(Q;X) = pos0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. Arke� na apode�xoume ìti(6:4) pos0(1)-Ind(Q;X) � pos0(0)-Ind(Q;X):'Estw pos0(0)-Ind(Q;X) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h sqèsh(6.4) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.4) e�nai alhj  gia k�jezeÔgo (QY ; Y ) me pos0(0)-Ind(QY ; Y ) < �. Ja apode�xoume ìti pos0(1)-Ind(Q;X) � �.Ef> ìson pos0(0)-Ind(Q;X) = �, up�rqei mia pos(0)-meg�lh b�sh B gia to Q sto X ètsi¸ste gia k�je U 2 B na èqoumepos0(0)-Ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,pos0(1)-Ind(Q \ BdX(U);BdX(U)) � pos0(0)-Ind(Q \ BdX(U);BdX(U)):Epeid  to uposÔnolo Q tou X e�nai kleistì, h B e�nai ep�sh mia pos(1)-meg�lh b�sh giato Q sto X. Sunep¸ , pos0(0)-Ind(Q;X) � �. �6.2.2 Par�deigma. 'Estw X = [�1; 1℄ kai Q = �� 12 ; 12�.H oikogèneia pou apotele�tai apì ìla ta sÔnola th morf  [�1; b) gia b > 0, (a; 1℄ giaa < 0 kai (a; b) e�nai b�sh gia mia topolog�a ep� tou X. ParathroÔme ìtipi(0)-Ind(Q;X) = posi(0)-Ind(Q;X) = 0 kaipi(1)-Ind(Q;X) = posi(1)-Ind(Q;X) > 0, i 2 f0; 1g:Epomènw oi anisìthte sthn Prìtash 6.2.1 den mporoÔn na antikatastajoÔn apì isìthte .6.2.3 Prìtash. 'Estw i 2 f0; 1g. Gia k�je uposÔnolo Q enì q¸rou X èqoumeInd(Q) � pi(1)-Ind(Q;X):

Diast�sei -sunart sei jèsew tou tÔpou Ind 99Apìdeixh. Ja apode�xoume ìti(6:5) Ind(Q) � p0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai an�logh. 'Estw p0(1)-Ind(Q;X) = � 2 O [ f�1;1g. E�n� = �1   � = 1, tìte h sqèsh (6.5) e�nai profan  . Upojètoume ìti � 2 O kai ìti hsqèsh (6.5) e�nai alhj  gia k�je zeÔgo (QY ; Y ) me p0(1)-Ind(QY ; Y ) < �. Ef> ìsonp0(1)-Ind(Q;X) = �, up�rqei mia p(1)-meg�lh b�sh B gia to Q sto X ètsi ¸ste gia k�jeU 2 B na èqoume p0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Epeid  to sÔnolo fQ \ U : U 2 Bg e�nai mia meg�lh b�sh gia ton upìqwro Q, gia naapode�xoume ìti Ind(Q) � � arke� na apode�xoume ìti Ind(BdQ(Q \ U)) < � gia k�jeU 2 B. Pr�gmati, ef> ìson BdQ(Q \ U) � Q \ BdX(U), apì thn upìjesh th epagwg  ,èqoumeInd(BdQ(Q \ U)) � Ind(Q \ BdX(U)) � p0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Sunep¸ , Ind(Q) � �. �6.2.4 Par�deigma. 'Estw X = fa; b; g kai Q = fa; bg. JewroÔme ep� tou X thntopolog�a � = f;; f g; fa; g; fb; g; Xg. Tìte,Ind(Q) = 0,pi(0)-Ind(Q;X) = posi(0)-Ind(Q;X) = 1 kaipi(1)-Ind(Q;X) = posi(1)-Ind(Q;X) = 1, i 2 f0; 1g.Epomènw h anisìthta sthn Prìtash 6.2.3 den mpore� na antikatastaje� apì isìthta.6.2.5 Prìtash. 'Estw i 2 f0; 1g. Gia k�je uposÔnolo Q enì q¸rou X èqoume(1) pi(0)-Ind(Q;X) � posi(0)-Ind(Q;X) kai(2) pi(1)-Ind(Q;X) � posi(1)-Ind(Q;X).Apìdeixh. (1) ApodeiknÔoume ìti(6:6) p0(0)-Ind(Q;X) � pos0(0)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0(0)-Ind(Q;X) = � 2 O[f�1;1g. E�n � = �1  � =1, tìte h sqèsh (6.6) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.6)e�nai alhj  gia k�je zeÔgo (QY ; Y ) me pos0(0)-Ind(QY ; Y ) < �. Ja apode�xoume ìti

100 Kef�laio 6p0(0)-Ind(Q;X) � �. Ef> ìson pos0(0)-Ind(Q;X) = �, up�rqei mia pos(0)-meg�lh b�shB gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumepos0(0)-Ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,p0(0)-Ind(Q \ BdX(U);BdX(U)) � pos0(0)-Ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia p(0)-meg�lh b�sh gia to Q sto X, p0(0)-Ind(Q;X) � �.(2) ApodeiknÔoume ìti(6:7) p0(1)-Ind(Q;X) � pos0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0(1)-Ind(Q;X) = � 2 O[f�1;1g. E�n � = �1  � =1, tìte h sqèsh (6.7) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.7)e�nai alhj  gia k�je zeÔgo (QY ; Y ) me pos0(1)-Ind(QY ; Y ) < �. Ja apode�xoume ìtip0(1)-Ind(Q;X) � �. Ef> ìson pos0(1)-Ind(Q;X) = �, up�rqei mia pos(1)-meg�lh b�shB gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumepos0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Apì thn upìjesh th epagwg  ,p0(1)-Ind(Q \ BdX(U);BdX(U)) � pos0(1)-Ind(Q \ BdX(U);BdX(U))kai epeid  h B e�nai ep�sh mia p(1)-meg�lh b�sh gia to Q sto X, p0(1)-Ind(Q;X) � �. �Apì ta parak�tw parade�gmata prokÔptei ìti oi anisìthte gia i = 0 sthn Prìtash6.2.5 den mporoÔn na antikatastajoÔn apì isìthte . Shmei¸noume ìti h per�ptwsh i = 1e�nai anoiktì prìblhma.6.2.6 Par�deigma. 'Estw X = fa; b; ; dg kai Q = fag. JewroÔme ep� tou X thntopolog�a � = f;; fbg; fa; bg; fb; ; dg; Xg. Tìte,p0(0)-Ind(Q;X) = p0(1)-Ind(Q;X) = 0,pos0(0)-Ind(Q;X) = pos0(1)-Ind(Q;X) = 1 kaipos1(0)-Ind(Q;X) = pos1(1)-Ind(Q;X) = 0.6.2.7 Par�deigma. 'Estw X = fa; b; g kai Q = fag. JewroÔme ep� tou X thn topolog�a� = f;; fa; bg; Xg. Tìte, p0(1)-Ind(Q;X) = 0 kai pos0(1)-Ind(Q;X) = 1.

Diast�sei -sunart sei jèsew tou tÔpou Ind 1016.2.8 Par�deigma. 'Estw X = fa; b; ; dg. JewroÔme ep� tou X thn topolog�a� = f;; fdg; fa; dg; fb; ; dg; Xg:Profan¸ , Ind(X) = 1 kai p0(1)-Ind(;; X) = pos0(1)-Ind(;; X) = 0. ParathroÔme ìtie�n Q = fa; b; g, tìtep1(0)-Ind(Q;X) = pos1(0)-Ind(Q;X) = 2 kaip1(1)-Ind(Q;X) = pos1(1)-Ind(Q;X) = 2.Ep�sh , e�n Q0 = fbg, tìte pos0(0)-Ind(Q0; X) = pos1(1)-Ind(Q0; X) = 0.6.2.9 Par�deigma. 'Estw X = fa; b; ; dg kai Q = fdg. JewroÔme ep� tou X thntopolog�a � = f;; fag; fa; bg; fa; dg; fa; b; g; fa; b; dg; Xg. Profan¸ , Ind(X) = 2 kaip0(1)-Ind(Q;X) = pos0(1)-Ind(Q;X) = 1.6.2.10 Parat rhsh. Oi sqèsei metaxÔ twn diast�sewn jèsew tou tÔpou Ind (blèpeti parap�nw prot�sei kai parade�gmata) sunoy�zontai sta parak�tw dÔo diagr�mmata,ìpou <<!>> shma�nei <<� >> kai <<9>> shma�nei ìti << genik� � >>.Ind(Q)��p0(1)-Ind(Q;X)�OO�

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Di�gramma 6.26.2.11 L mma. 'Estw X klhronomik� fusikì q¸ro kai Q � X. Gia k�je anoiktìuposÔnolo V Q tou Q kai gia k�je anoiktì uposÔnolo U tou X me V Q � U up�rqeianoiktì uposÔnolo V tou X ètsi ¸ste V � U , V Q = Q\V kai BdQ(V Q) = Q\BdX(V ).

102 Kef�laio 6Apìdeixh. Ta sÔnola V Q kai Q n ClQ(V Q) e�nai diaqwrismèna sto X. Epeid  o X e�naiklhronomik� fusikì , up�rqei anoiktì uposÔnolo V0 tou X tètoio ¸steV Q � V0 kai ClX(V0) \ (Q n ClQ(V Q)) = ;:'Estw V1 anoiktì uposÔnolo tou X tètoio ¸ste V Q = Q \ V1. JètoumeV = V0 \ V1 \ U:Tìte, V � U , V Q = Q \ V � V kai ClX(V ) \ (Q n ClQ(V Q)) = ;. 'Ara,Q \ ClX(V ) � ClQ(V Q):Ep�sh , ClQ(V Q) = Q \ ClX(V Q) � Q \ ClX(V ):Sunep¸ , ClQ(V Q) = Q \ ClX(V ) kai epomènw BdQ(V Q) = ClQ(V Q) n V Q = (Q \ ClX(V )) n (Q \ V ) = Q \ BdX(V ):'Ara, V � U , V Q = Q \ V kai BdQ(V Q) = Q \ BdX(V ). �6.2.12 Prìtash. 'Estw X klhronomik� fusikì q¸ro (dhlad  k�je upìqwro tou Xe�nai fusikì ) kai Q � X. Tìte,pos1(1)-Ind(Q;X) = Ind(Q):Apìdeixh. Apì ti Prot�sei 6.2.3 kai 6.2.5(2), arke� na apode�xoume ìti(6:8) pos1(1)-Ind(Q;X) � Ind(Q):'Estw Ind(Q) = � 2 O [ f�1;1g. E�n � = �1   � = 1, tìte h sqèsh (6.8) e�naiprofan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.8) e�nai alhj  gia k�je uposÔnoloM tou X me Ind(M) < �. 'Estw (FQ; U) èna zeÔgo apì uposÔnola tou X, ìpou to FQe�nai kleistì uposÔnolo tou Q, to U e�nai anoiktì uposÔnolo tou X kai FQ � U . Ef>ìson Ind(Q) = �, up�rqei anoiktì uposÔnolo V Q tou Q tètoio ¸ste FQ � V Q � Q \ Ukai Ind(BdQ(V Q)) < �. Apì to L mma 6.2.11, up�rqei anoiktì uposÔnolo V tou X ètsi¸ste V � U , V Q = Q\ V kai BdQ(V Q) = Q\BdX(V ). Apì thn upìjesh th epagwg  ,èqoume pos1(1)-Ind(Q \ BdX(V ); X) � Ind(BdQ(V Q)) < �:Sunep¸ , pos1(1)-Ind(Q;X) � �. �

Diast�sei -sunart sei jèsew tou tÔpou Ind 1036.2.13 Pìrisma. 'Estw X klhronomik� fusikì q¸ro (dhlad  k�je upìqwro tou Xe�nai fusikì ) kai Q � X. Tìte, p1(1)-Ind(Q;X) = Ind(Q).Apìdeixh. Apì ti Prot�sei 6.2.3, 6.2.5(2) kai 6.2.12, èqoumeInd(Q) � p1(1)-Ind(Q;X) � pos1(1)-Ind(Q;X) = Ind(Q):Sunep¸ , p1(1)-Ind(Q;X) = Ind(Q). �6.2.14 Pìrisma. 'Estw X klhronomik� fusikì q¸ro (dhlad  k�je upìqwro tou Xe�nai fusikì ) kai Q � X. Tìte, pos1(0)-Ind(Q;X) � Ind(Q).Apìdeixh. Apì ti Prot�sei 6.2.1(2) kai 6.2.12, èqoumepos1(0)-Ind(Q;X) � pos1(1)-Ind(Q;X) = Ind(Q):Sunep¸ , pos1(0)-Ind(Q;X) � Ind(Q). �6.2.15 Prìtash. 'Estw i 2 f0; 1g. Gia k�je uposÔnolo Q enì T1-q¸rou X èqoume(1) pi-ind(Q;X) � pi(0)-Ind(Q;X) kai(2) posi-ind(Q;X) � posi(0)-Ind(Q;X).Apìdeixh. ProkÔptei �mesa apì to gegonì ìti s> ènan T1-q¸ro k�je monosÔnolo e�naikleistì. �6.3 Jewr mata Upoq¸rou6.3.1 Prìtash. 'Estw i 2 f0; 1g kai Q;K dÔo uposÔnola enì q¸rou X me K � Q.Tìte,(a) pi(0)-Ind(K;X) � pi(0)-Ind(Q;X) kai(b) posi(0)-Ind(K;X) � posi(0)-Ind(Q;X).Apìdeixh. (a) ApodeiknÔoume thn anisìthta(6:9) p0(0)-Ind(K;X) � p0(0)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw p0(0)-Ind(Q;X) = �, ìpou � 2 O [ f�1;1g. E�n� = �1   � =1, tìte h anisìthta (6.9) e�nai profan  . Upojètoume ìti � 2 O kai ìti hanisìthta (6.9) e�nai alhj  gia k�je K � Q � X me p0(0)-Ind(Q;X) < �. Up�rqei miap(0)-meg�lh b�sh B gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep0(0)-Ind(Q \ BdX(U);BdX(U)) < �:

104 Kef�laio 6Ef> ìson K \ BdX(U) � Q \ BdX(U), apì thn upìjesh th epagwg  ,p0(0)-Ind(K \ BdX(U);BdX(U)) � p0(0)-Ind(Q \ BdX(U);BdX(U)) < �kai epeid  h B e�nai mia p(0)-meg�lh b�sh gia to K sto X, p0(0)-Ind(K;X) � �.(b) ApodeiknÔoume thn anisìthta(6:10) pos0(0)-Ind(K;X) � pos0(0)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0(0)-Ind(Q;X) = �, ìpou � 2 O [ f�1;1g.E�n � = �1   � = 1, tìte h anisìthta (6.10) e�nai profan  . Upojètoume ìti � 2 Okai ìti h anisìthta (6.10) e�nai alhj  gia k�je K � Q � X me pos0(0)-Ind(Q;X) < �.Up�rqei mia pos(0)-meg�lh b�sh B gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumepos0(0)-Ind(Q \ BdX(U);BdX(U)) < �:Ef> ìson K \ BdX(U) � Q \ BdX(U), apì thn upìjesh th epagwg  ,pos0(0)-Ind(K \ BdX(U);BdX(U)) � pos0(0)-Ind(Q \ BdX(U);BdX(U)) < �kai epeid  h B e�nai mia pos(0)-meg�lh b�sh gia to K sto X, pos0(0)-Ind(K;X) � �. �6.3.2 Prìtash. 'Estw i 2 f0; 1g, Y èna kleistì upìqwro enì q¸rou X kai Q � Y .Tìte,(a) pi(0)-Ind(Q; Y ) � pi(0)-Ind(Q;X) kai(b) posi(0)-Ind(Q; Y ) � posi(0)-Ind(Q;X).Apìdeixh. (a) ApodeiknÔoume thn anisìthta(6:11) p1(0)-Ind(Q; Y ) � p1(0)-Ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw p1(0)-Ind(Q;X) = �, ìpou � 2 O [ f�1;1g. E�n� = �1   � = 1, tìte h sqèsh (6.11) e�nai profan  . Upojètoume ìti � 2 O kai ìtih sqèsh (6.11) e�nai alhj  gia k�je Q � Y � X, ìpou to Y e�nai kleistì sto X, mep1(0)-Ind(Q;X) < �. Up�rqei mia p(0)-meg�lh b�sh B gia to Q sto X ètsi ¸ste giak�je U 2 B na èqoume p1(0)-Ind(Q \ BdX(U); X) < �:Ef> ìson BdY (U \ Y ) � Y \ BdX(U) � BdX(U), apì thn Prìtash 6.3.1(a),p1(0)-Ind(Q \ BdY (U \ Y ); X) � p1(0)-Ind(Q \ BdX(U); X) < �:

Diast�sei -sunart sei jèsew tou tÔpou Ind 105Ep�sh , apì thn upìjesh th epagwg  ,p1(0)-Ind(Q \ BdY (U \ Y ); Y ) � p1(0)-Ind(Q \ BdY (U \ Y ); X) < �:Ef> ìson o upìqwro Y tou X e�nai kleistì , to sÔnolo fU \ Y : U 2 Bg e�nai miap(0)-meg�lh b�sh gia to Q sto Y . Sunep¸ , p1(0)-Ind(Q; Y ) � �.(b) ApodeiknÔoume thn anisìthta(6:12) pos1(0)-Ind(Q; Y ) � pos1(0)-Ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw pos1(0)-Ind(Q;X) = �, ìpou � 2 O [ f�1;1g.E�n � = �1   � = 1, tìte h sqèsh (6.12) e�nai profan  . Upojètoume ìti � 2 O kaiìti h sqèsh (6.12) e�nai alhj  gia k�je Q � Y � X, ìpou to Y e�nai kleistì sto X, mepos1(0)-Ind(Q;X) < �. Up�rqei mia pos(0)-meg�lh b�sh B gia to Q sto X ètsi ¸ste giak�je U 2 B na èqoume pos1(0)-Ind(Q \ BdX(U); X) < �:Ef> ìson BdY (U \ Y ) � Y \ BdX(U) � BdX(U), apì thn Prìtash 6.3.1(b),pos1(0)-Ind(Q \ BdY (U \ Y ); X) � pos1(0)-Ind(Q \ BdX(U); X) < �:Ep�sh , apì thn upìjesh th epagwg  ,pos1(0)-Ind(Q \ BdY (U \ Y ); Y ) � pos1(0)-Ind(Q \ BdY (U \ Y ); X) < �:Ef> ìson o upìqwro Y tou X e�nai kleistì , to sÔnolo fU \ Y : U 2 Bg e�nai miapos(0)-meg�lh b�sh gia to Q sto Y . Sunep¸ , pos1(0)-Ind(Q; Y ) � �. �6.3.3 Prìtash. 'Estw i 2 f0; 1g kai Q;K dÔo uposÔnola enì q¸rou X me K � Q. E�nto uposÔnolo K e�nai kleistì sto Q, tìte(a) pi(1)-Ind(K;X) � pi(1)-Ind(Q;X) kai(b) posi(1)-Ind(K;X) � posi(1)-Ind(Q;X).Apìdeixh. (a) ApodeiknÔoume thn anisìthta(6:13) p1(1)-Ind(K;X) � p1(1)-Ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw p1(1)-Ind(Q;X) = �, ìpou � 2 O [ f�1;1g. E�n� = �1   � =1, tìte h anisìthta (6.13) e�nai profan  . Upojètoume ìti � 2 O kai ìtih anisìthta (6.13) e�nai alhj  gia k�je K � Q � X, ìpou to K e�nai kleistì sto Q,

106 Kef�laio 6me p1(1)-Ind(Q;X) < �. Up�rqei mia p(1)-meg�lh b�sh B gia to Q sto X ètsi ¸ste giak�je U 2 B na èqoume p1(1)-Ind(Q \ BdX(U); X) < �:Ef> ìson to uposÔnolo K \BdX(U) e�nai kleistì sto Q\BdX(U), apì thn upìjesh th epagwg  , p1(1)-Ind(K \ BdX(U); X) � p1(1)-Ind(Q \ BdX(U); X) < �:Epiplèon, ef> ìson to K e�nai kleistì sto Q, h oikogèneia B e�nai mia p(1)-meg�lh b�shgia to K sto X. Sunep¸ , p1(1)-Ind(K;X) � �.(b) ApodeiknÔoume thn anisìthta(6:14) pos1(1)-Ind(K;X) � pos1(1)-Ind(Q;X):H per�ptwsh i = 0 e�nai ìmoia. 'Estw pos1(1)-Ind(Q;X) = �, ìpou � 2 O [ f�1;1g.E�n � = �1   � =1, tìte h anisìthta (6.14) e�nai profan  . Upojètoume ìti � 2 O kaiìti h anisìthta (6.14) e�nai alhj  gia k�je K � Q � X, ìpou to K e�nai kleistì sto Q,me pos1(1)-Ind(Q;X) < �. Up�rqei mia pos(1)-meg�lh b�sh B gia to Q sto X ètsi ¸stegia k�je U 2 B na èqoume pos1(1)-Ind(Q \ BdX(U); X) < �:Ef> ìson to uposÔnolo K \BdX(U) e�nai kleistì sto Q\BdX(U), apì thn upìjesh th epagwg  , pos1(1)-Ind(K \ BdX(U); X) � pos1(1)-Ind(Q \ BdX(U); X) < �:Epiplèon, ef> ìson to K e�nai kleistì sto Q, h oikogèneia B e�nai mia pos(1)-meg�lh b�shgia to K sto X. Sunep¸ , pos1(1)-Ind(K;X) � �. �6.3.4 Prìtash. 'Estw i 2 f0; 1g, Y èna upìqwro enì q¸rou X kai Q � Y . Tìte,(a) pi(1)-Ind(Q; Y ) � pi(1)-Ind(Q;X) kai(b) posi(1)-Ind(Q; Y ) � posi(1)-Ind(Q;X).Apìdeixh. (a) ApodeiknÔoume thn anisìthta(6:15) p0(1)-Ind(Q; Y ) � p0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw p0(1)-Ind(Q;X) = � 2 O[f�1;1g. E�n � = �1  � =1, tìte h sqèsh (6.15) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.15)

Diast�sei -sunart sei jèsew tou tÔpou Ind 107e�nai alhj  gia k�je Q � Y � X me p0(1)-Ind(Q;X) < �. Up�rqei mia p(1)-meg�lh b�shB gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumep0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Ef> ìson to uposÔnolo Q\BdY (U \ Y ) e�nai kleistì sto Q\BdX(U), apì thn Prìtash6.3.3(a),p0(1)-Ind(Q \ BdY (U \ Y );BdX(U)) � p0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Ep�sh , apì thn upìjesh th epagwg  ,p0(1)-Ind(Q \ BdY (U \ Y );BdY (U \ Y )) � p0(1)-Ind(Q \ BdY (U \ Y );BdX(U)) < �kai epeid  to sÔnolo fU \ Y : U 2 Bg e�nai mia p(1)-meg�lh b�sh gia to Q sto Y , èqoumep0(1)-Ind(Q; Y ) � �.(b) ApodeiknÔoume thn anisìthta(6:16) pos0(1)-Ind(Q; Y ) � pos0(1)-Ind(Q;X):H per�ptwsh i = 1 e�nai ìmoia. 'Estw pos0(1)-Ind(Q;X) = � 2 O[f�1;1g. E�n � = �1  � =1, tìte h sqèsh (6.16) e�nai profan  . Upojètoume ìti � 2 O kai ìti h sqèsh (6.16)e�nai alhj  gia k�je Q � Y � X me pos0(1)-Ind(Q;X) < �. Up�rqei mia pos(1)-meg�lhb�sh B gia to Q sto X ètsi ¸ste gia k�je U 2 B na èqoumepos0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Ef> ìson to uposÔnolo Q\BdY (U \ Y ) e�nai kleistì sto Q\BdX(U), apì thn Prìtash6.3.3(b),pos0(1)-Ind(Q \ BdY (U \ Y );BdX(U)) � pos0(1)-Ind(Q \ BdX(U);BdX(U)) < �:Ep�sh , apì thn upìjesh th epagwg  ,pos0(1)-Ind(Q\BdY (U \ Y );BdY (U \ Y )) � pos0(1)-Ind(Q\BdY (U \Y );BdX(U)) < �kai epeid  to sÔnolo fU \ Y : U 2 Bg e�nai mia p(1)-meg�lh b�sh gia to Q sto Y , èqoumepos0(1)-Ind(Q; Y ) � �. �

108 Kef�laio 66.4 Jewr mata DiaqwrismoÔSthn enìthta aut  jewroÔme èna stajerì q¸ro X, èna uposÔnolo Q tou X kai ènadiataktikì arijmì �.6.4.1 Prìtash. (a) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist� uposÔnolatou X, ìpou A � Q, up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnolaA kai B ètsi ¸ste p0(0)-Ind(Q \ L; L) < �, tìte p0(0)-Ind(Q;X) � �.(b) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist� uposÔnola tou X, ìpouA � Q, up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnola A kai Bètsi ¸ste p1(0)-Ind(Q \ L;X) < �, tìte p1(0)-Ind(Q;X) � �.Apìdeixh. (a) 'Estw (F; U) zeÔgo apì uposÔnola tou X, ìpou to F e�nai kleistìuposÔnolo tou X, to U e�nai anoiktì uposÔnolo tou X kai F � Q\ U . Apì th upìjesh,up�rqei èna uposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola F kai X n U ètsi ¸step0(0)-Ind(Q \ L; L) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) F � V , X n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoume F � V � X nW � U . 'Ara, F � Q \ V � Q \ U .Ep�sh , apì ti sqèsei (2) kai (3), èqoumeBdX(V ) = ClX(V ) \ ClX(X n V ) = ClX(V ) \ (X n V )� ClX(X nW ) \ (X n V ) = (X nW ) \ (X n V )= X n (V [W ) = L:Ef> ìson p0(0)-Ind(Q \ L; L) < � kai BdX(V ) � L, apì thn Prìtash 6.3.1(a),p0(0)-Ind(Q \ BdX(V ); L) � p0(0)-Ind(Q \ L; L) < �:Ep�sh , ef> ìson to BdX(V ) e�nai kleistì uposÔnolo tou L, apì thn Prìtash 6.3.2(a),p0(0)-Ind(Q \ BdX(V );BdX(V )) � p0(0)-Ind(Q \ BdX(V ); L):Sunep¸ , p0(0)-Ind(Q \ BdX(V );BdX(V )) < � kai epomènw p0(0)-Ind(Q;X) � �.

Diast�sei -sunart sei jèsew tou tÔpou Ind 109(b) 'Estw (F; U) zeÔgo apì uposÔnola tou X, ìpou to F e�nai kleistì uposÔnolo touX, to U e�nai anoiktì uposÔnolo tou X kai F � Q \ U . Apì th upìjesh, up�rqei ènauposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola F kai X n U ètsi ¸step1(0)-Ind(Q \ L;X) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) F � V , X n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoume F � V � X nW � U . 'Ara, F � Q \ V � Q \ U .Ep�sh , apì ti sqèsei (2) kai (3), èqoume BdX(V ) � L. Ef> ìson p1(0)-Ind(Q\L;X) < �kai BdX(V ) � L, apì thn Prìtash 6.3.1(a),p1(0)-Ind(Q \ BdX(V ); X) � p1(0)-Ind(Q \ L;X) < �:Sunep¸ , p1(0)-Ind(Q \ BdX(V ); X) < � kai epomènw p1(0)-Ind(Q;X) � �. �6.4.2 Prìtash. (a) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist� uposÔnolatou X, ìpou A � Q, up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnolaA kai B ètsi ¸ste pos0(0)-Ind(Q \ L; L) < �, tìte pos0(0)-Ind(Q;X) � �.(b) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist� uposÔnola tou X, ìpouA � Q, up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnola A kai Bètsi ¸ste pos1(0)-Ind(Q \ L;X) < �, tìte pos1(0)-Ind(Q;X) � �.Apìdeixh. (a) 'Estw (F; U) zeÔgo apì uposÔnola tou X, ìpou to F e�nai kleistìuposÔnolo tou X, to U e�nai anoiktì uposÔnolo tou X kai F � Q\ U . Apì th upìjesh,up�rqei èna uposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola F kai X n U ètsi ¸stepos0(0)-Ind(Q \ L; L) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) F � V , X n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoume F � V � X nW � U . Ep�sh , apì ti sqèsei (2) kai(3), èqoume BdX(V ) = ClX(V ) \ ClX(X n V ) = ClX(V ) \ (X n V )

110 Kef�laio 6� ClX(X nW ) \ (X n V ) = (X nW ) \ (X n V )= X n (V [W ) = L:Ef> ìson pos0(0)-Ind(Q \ L; L) < � kai BdX(V ) � L, apì thn Prìtash 6.3.1(b),pos0(0)-Ind(Q \ BdX(V ); L) � p0(0)-Ind(Q \ L; L) < �:Ep�sh , ef> ìson to BdX(V ) e�nai kleistì uposÔnolo tou L, apì thn Prìtash 6.3.2(b),pos0(0)-Ind(Q \ BdX(V );BdX(V )) � pos0(0)-Ind(Q \ BdX(V ); L):Sunep¸ , pos0(0)-Ind(Q \ BdX(V );BdX(V )) < � kai epomènw pos0(0)-Ind(Q;X) � �.(b) 'Estw (F; U) zeÔgo apì uposÔnola tou X, ìpou to F e�nai kleistì uposÔnolo touX, to U e�nai anoiktì uposÔnolo tou X kai F � Q \ U . Apì th upìjesh, up�rqei ènauposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola F kai X n U ètsi ¸stepos1(0)-Ind(Q \ L;X) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) F � V , X n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoume F � V � X nW � U . Ep�sh , apì ti sqèsei (2) kai(3), èqoume BdX(V ) � L. Ef> ìson pos1(0)-Ind(Q\L;X) < � kai BdX(V ) � L, apì thnPrìtash 6.3.1(b),pos1(0)-Ind(Q \ BdX(V ); X) � p1(0)-Ind(Q \ L;X) < �:Sunep¸ , pos1(0)-Ind(Q \ BdX(V ); X) < � kai epomènw pos1(0)-Ind(Q;X) � �. �Sthn parak�tw prìtash apodeiknÔetai to ant�strofo th Prìtash 6.4.2 gia fusikoÔ q¸rou .6.4.3 Prìtash. (a) E�n pos0(0)-Ind(Q;X) � �, tìte gia k�je zeÔgo (A;B) apì xènametaxÔ tou kleist� uposÔnola tou X, ìpou A � Q, up�rqei èna uposÔnolo L tou q¸rouX to opo�o diaqwr�zei ta sÔnola A kai B ètsi ¸ste pos0(0)-Ind(Q \ L; L) < �.(b) E�n pos1(0)-Ind(Q;X) � �, tìte gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist�uposÔnola tou X, ìpou A � Q, up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zeita sÔnola A kai B ètsi ¸ste pos1(0)-Ind(Q \ L;X) < �.

Diast�sei -sunart sei jèsew tou tÔpou Ind 111Apìdeixh. (a) 'Estw (A;B) zeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X, ìpouA � Q. Tìte, to X nB e�nai anoiktì uposÔnolo tou q¸rou X kai A � X nB. Epomènw ,epeid  o q¸ro X e�nai fusikì , up�rqei anoiktì uposÔnolo U tou X ètsi ¸steA � U � ClX(U) � X nB:Ep�sh , epeid  pos0(0)-Ind(Q;X) � �, up�rqei anoiktì uposÔnolo V tou X ètsi ¸steA � V � U � ClX(U) � X nBkai pos0(0)-Ind(Q \ BdX(V );BdX(V )) < �:Ef> ìson to sÔnolo BdX(V ) diaqwr�zei ta A kai B, to zhtoÔmeno uposÔnolo L tou Xe�nai to BdX(V ).(b) 'Estw (A;B) zeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X, ìpou A � Q.Tìte, to X nB e�nai anoiktì uposÔnolo tou q¸rou X kai A � X n B. Epomènw , epeid o q¸ro X e�nai fusikì , up�rqei anoiktì uposÔnolo U tou X ètsi ¸steA � U � ClX(U) � X nB:Ep�sh , epeid  pos1(0)-Ind(Q;X) � �, up�rqei anoiktì uposÔnolo V tou X ètsi ¸steA � V � U � ClX(U) � X nBkai pos1(0)-Ind(Q \ BdX(V ); X) < �:Ef> ìson to sÔnolo BdX(V ) diaqwr�zei ta A kai B, to zhtoÔmeno uposÔnolo L tou Xe�nai to BdX(V ). �6.4.4 Prìtash. (a) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist� uposÔnolatou upoq¸rou Q up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnola Akai B ètsi ¸ste p0(1)-Ind(Q \ L; L) < �, tìte p0(1)-Ind(Q;X) � �.(b) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist� uposÔnola tou upoq¸rou Qup�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnola A kai B ètsi ¸step1(1)-Ind(Q \ L;X) < �, tìte p1(1)-Ind(Q;X) � �.Apìdeixh. (a) 'Estw (FQ; U) zeÔgo apì uposÔnola tou X, ìpou to FQ e�nai kleistìuposÔnolo tou Q, to U e�nai anoiktì uposÔnolo tou X kai FQ � U . Apì th upìjesh,up�rqei èna uposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola FQ kai Q n U ètsi ¸step0(1)-Ind(Q \ L; L) < �:

112 Kef�laio 6'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) FQ � V , Q n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoumeFQ � Q \ V � Q \ (X nW ) � Q \ U:Ep�sh , apì ti sqèsei (2) kai (3), èqoume BdX(V ) � L. Ef> ìson p0(1)-Ind(Q\L; L) < �kai to Q \ BdX(V ) e�nai kleistì uposÔnolo tou Q \ L, apì ti Prot�sei 6.3.3(a) kai6.3.4(a), èqoume p0(1)-Ind(Q \ BdX(V );BdX(V )) < �:Sunep¸ , p0(1)-Ind(Q;X) � �.(b) 'Estw (FQ; U) zeÔgo apì uposÔnola tou X, ìpou to FQ e�nai kleistì uposÔnolotou Q, to U e�nai anoiktì uposÔnolo tou X kai FQ � U . Apì th upìjesh, up�rqei ènauposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola FQ kai Q n U ètsi ¸step1(1)-Ind(Q \ L;X) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) FQ � V , Q n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoumeFQ � Q \ V � Q \ (X nW ) � Q \ U:Ep�sh , apì ti sqèsei (2) kai (3), èqoume BdX(V ) � L. Ef> ìson p1(1)-Ind(Q\L;X) < �kai to Q\BdX(V ) e�nai kleistì uposÔnolo tou Q\L, apì thn Prot�sh 6.3.3(a), èqoumep1(1)-Ind(Q \ BdX(V ); X) < �:Sunep¸ , p1(1)-Ind(Q;X) � �. �6.4.5 Prìtash. (a) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou uposÔnola touX, ìpou to A e�nai kleistì uposÔnolo tou Q kai to B e�nai kleistì uposÔnolo tou X,up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnola A kai B ètsi ¸stepos0(1)-Ind(Q \ L; L) < �, tìte pos0(1)-Ind(Q;X) � �.(b) E�n gia k�je zeÔgo (A;B) apì xèna metaxÔ tou uposÔnola tou X, ìpou to A e�nai

Diast�sei -sunart sei jèsew tou tÔpou Ind 113kleistì uposÔnolo tou Q kai to B e�nai kleistì uposÔnolo tou X, up�rqei èna uposÔnoloL tou q¸rou X to opo�o diaqwr�zei ta sÔnola A kai B ètsi ¸ste pos1(1)-Ind(Q\L;X) <�, tìte pos1(1)-Ind(Q;X) � �.Apìdeixh. (a) 'Estw (FQ; U) zeÔgo apì uposÔnola tou X, ìpou to FQ e�nai kleistìuposÔnolo tou Q, to U e�nai anoiktì uposÔnolo tou X kai FQ � U . Apì th upìjesh,up�rqei èna uposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola FQ kai X n U ètsi ¸stepos0(1)-Ind(Q \ L; L) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) FQ � V , X n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoume FQ � V � X nW � U . Ep�sh , apì ti sqèsei (2)kai (3), èqoume BdX(V ) � L. Ef> ìson pos0(1)-Ind(Q \ L; L) < � kai to Q \ BdX(V )e�nai kleistì uposÔnolo tou Q \ L, apì ti Prot�sei 6.3.3(b) kai 6.3.4(b), èqoumepos0(1)-Ind(Q \ BdX(V );BdX(V )) < �:Sunep¸ , pos0(1)-Ind(Q;X) � �.(b) 'Estw (FQ; U) zeÔgo apì uposÔnola tou X, ìpou to FQ e�nai kleistì uposÔnolotou Q, to U e�nai anoiktì uposÔnolo tou X kai FQ � U . Apì th upìjesh, up�rqei ènauposÔnolo L tou X to opo�o diaqwr�zei ta sÔnola FQ kai X n U ètsi ¸stepos1(1)-Ind(Q \ L;X) < �:'Estw V kai W anoikt� uposÔnola tou X ètsi ¸ste:(1) FQ � V , X n U � W ,(2) V \W = ; kai(3) X n L = V [W .Apì ti sqèsei (1) kai (2), èqoume FQ � V � X nW � U . Ep�sh , apì ti sqèsei (2)kai (3), èqoume BdX(V ) � L. Ef> ìson pos1(1)-Ind(Q \ L;X) < � kai to Q \ BdX(V )e�nai kleistì uposÔnolo tou Q \ L, apì thn Prot�sh 6.3.3(b), èqoumepos1(1)-Ind(Q \ BdX(V ); X) < �:Sunep¸ , pos1(1)-Ind(Q;X) � �. �

114 Kef�laio 66.5 Jewr mata Ajro�smato 6.5.1 L mma. 'Estw Q kleistì uposÔnolo enì klhronomik� fusikoÔ q¸rouX kai � 2 O.(a) E�n p1(0)-Ind(Q;X) � �, tìte gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist�uposÔnola tou X up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnolaA kai B ètsi ¸ste p1(0)-Ind(Q \ L;X) < �.(b) E�n pos1(0)-Ind(Q;X) � �, tìte gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist�uposÔnola tou X up�rqei èna uposÔnolo L tou q¸rou X to opo�o diaqwr�zei ta sÔnolaA kai B ètsi ¸ste pos1(0)-Ind(Q \ L;X) < �.Apìdeixh. (a) 'Estw (A;B) zeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X. Ef>ìson o q¸ro X e�nai fusikì , up�rqoun dÔo anoikt� uposÔnola U1; U2 tou X ètsi ¸steA � U1, B � U2 kai ClX(U1) \ ClX(U2) = ;. Ef> ìson o Q e�nai fusikì , up�rqei ènaanoiktì uposÔnolo U tou X ètsi ¸steQ \ ClX(U1) � Q \ U � ClQ(Q \ U) � Q n ClX(U2):Ep�sh , epeid  p1(0)-Ind(Q;X) � �, up�rqei èna anoiktì uposÔnolo V tou X ètsi ¸steQ \ ClX(U1) � Q \ V � Q \ U � ClQ(Q \ U) � Q n ClX(U2)kai p1(0)-Ind(Q \ BdX(V ); X) < �:Apì thn Prìtash 6.3.1(a) èqoumep1(0)-Ind(BdQ(Q \ V ); X) � p1(0)-Ind(Q \ BdX(V ); X):Ep�sh , to sÔnolo BdQ(Q \ V ) diaqwr�zei ta Q \ ClX(U1) kai Q \ ClX(U2) sto Q. Apìto L mma 1.2.9 tou [19℄ (blèpe ep�sh Parat rhsh 1.2.10 tou [19℄), up�rqei èna uposÔnoloL tou X to opo�o diaqwr�zei ta A kai B ètsi ¸ste Q \ L = BdQ(Q \ V ). Sunep¸ ,p1(0)-Ind(Q \ L;X) < �.(b) 'Estw (A;B) zeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X. Ef> ìson oq¸ro X e�nai fusikì , up�rqoun dÔo anoikt� uposÔnola U1; U2 tou X ètsi ¸ste A � U1,B � U2 kai ClX(U1) \ ClX(U2) = ;. Ef> ìson o Q e�nai fusikì , up�rqei èna anoiktìuposÔnolo U tou X ètsi ¸steQ \ ClX(U1) � Q \ U � ClQ(Q \ U) � Q n ClX(U2):Ep�sh , epeid  pos1(0)-Ind(Q;X) � �, up�rqei èna anoiktì uposÔnolo V tou X ètsi ¸steQ \ ClX(U1) � Q \ V � Q \ U � ClQ(Q \ U) � Q n ClX(U2)

Diast�sei -sunart sei jèsew tou tÔpou Ind 115kai pos1(0)-Ind(Q \ BdX(V ); X) < �:Apì thn Prìtash 6.3.1(b) èqoumepos1(0)-Ind(BdQ(Q \ V ); X) � pos1(0)-Ind(Q \ BdX(V ); X):Ep�sh , to sÔnolo BdQ(Q \ V ) diaqwr�zei ta Q \ ClX(U1) kai Q \ ClX(U2) sto Q. Apìto L mma 1.2.9 tou [19℄ (blèpe ep�sh Parat rhsh 1.2.10 tou [19℄), up�rqei èna uposÔnoloL tou X to opo�o diaqwr�zei ta A kai B ètsi ¸ste Q \ L = BdQ(Q \ V ). Sunep¸ ,pos1(0)-Ind(Q \ L;X) < �. �6.5.2 Prìtash. 'Estw Q1 kai Q2 dÔo upìqwroi enì klhronomik� fusikoÔ q¸rou X.E�n to uposÔnolo Q1 tou X e�nai kleistì, tìte(1) p1(0)-Ind(Q1 [Q2; X) � p1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X) + 1 kai(2) pos1(0)-Ind(Q1 [Q2; X) � pos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X) + 1.Apìdeixh. (1) ApodeiknÔoume th sqèsh(6:17) p1(0)-Ind(Q1 [Q2; X) � p1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X) + 1me epagwg  sto �, ìpou � = p1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X). E�n � = �1, tìtep1(0)-Ind(Q1; X) = p1(0)-Ind(Q2; X) = �1 pou shma�nei ìti Q1 [ Q2 = ; kai sunep¸ h(6.17) e�nai alhj  . Upojètoume ìti gia k�je klhronomik� fusikì q¸ro X kai k�je dÔouposÔnola Q1; Q2 tou X h sqèsh (6.17) e�nai alhj  e�np1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X) < �;ìpou � e�nai èna stajerì diataktikì arijmì . ApodeiknÔoume th sqèsh (6.17) gia thnper�ptwsh p1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X) = �. 'Estwp1(0)-Ind(Q1; X) = �1kai p1(0)-Ind(Q1; X) = �2;ìpou �1; �2 2 O [ f�1g. E�n �1 = �1   �2 = �1, tìte Q1 = ;   Q2 = ;, ant�stoiqa kaih sqèsh (6.17) e�nai alhj  . Upojètoume ìti �1; �2 2 O.Apì thn Prìtah 6.4.1(b), arke� na apode�xoume ìti gia k�je zeÔgo (A;B) apì xènametaxÔ tou kleist� uposÔnola tou X, ìpou A � Q, up�rqei èna uposÔnolo L tou X

116 Kef�laio 6to opo�o diaqwr�zei ta A kai B ètsi ¸ste p1(0)-Ind(Q \ L;X) < �. 'Estw (A;B) ènazeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X. Apì to L mma 6.5.1(a), up�rqeièna uposÔnolo L tou X to opo�o diaqwr�zei ta A kai B ètsi ¸step1(0)-Ind(Q1 \ L;X) < �1:Ep�sh , apì thn Prìtash 6.3.1(a),p1(0)-Ind(Q2 \ L;X) � p1(0)-Ind(Q2; X) = �2:'Ara, p1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X) < �1(+)�2 = �:Ef> ìson (Q1 [Q2) \ L = (Q1 \ L) [ (Q2 \ L), apì thn upìjesh th epagwg  ,p1(0)-Ind((Q1 [Q2) \ L;X) < � + 1:Sunep¸ ,p1(0)-Ind(Q1 [Q2; X) � � + 1 = p1(0)-Ind(Q1; X)(+)p1(0)-Ind(Q2; X) + 1:(2) ApodeiknÔoume th sqèsh(6:18) pos1(0)-Ind(Q1 [Q2; X) � pos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X) + 1me epagwg  sto �, ìpou � = pos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X). E�n � = �1,tìte pos1(0)-Ind(Q1; X) = pos1(0)-Ind(Q2; X) = �1 pou shma�nei ìti Q1 [ Q2 = ; kaisunep¸ h (6.18) e�nai alhj  . Upojètoume ìti gia k�je klhronomik� fusikì q¸ro X kaik�je dÔo uposÔnola Q1; Q2 tou X h sqèsh (6.18) e�nai alhj  e�npos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X) < �;ìpou � e�nai èna stajerì diataktikì arijmì . ApodeiknÔoume th sqèsh (6.18) gia thnper�ptwsh pos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X) = �. 'Estwpos1(0)-Ind(Q1; X) = �1kai pos1(0)-Ind(Q1; X) = �2;ìpou �1; �2 2 O [ f�1g. E�n �1 = �1   �2 = �1, tìte Q1 = ;   Q2 = ;, ant�stoiqa kaih sqèsh (6.18) e�nai alhj  . Upojètoume ìti �1; �2 2 O.

Diast�sei -sunart sei jèsew tou tÔpou Ind 117Apì thn Prìtah 6.4.2(b), arke� na apode�xoume ìti gia k�je zeÔgo (A;B) apì xènametaxÔ tou kleist� uposÔnola tou X, ìpou A � Q, up�rqei èna uposÔnolo L tou Xto opo�o diaqwr�zei ta A kai B ètsi ¸ste pos1(0)-Ind(Q \ L;X) < �. 'Estw (A;B) ènazeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X. Apì to L mma 6.5.1(b), up�rqeièna uposÔnolo L tou X to opo�o diaqwr�zei ta A kai B ètsi ¸stepos1(0)-Ind(Q1 \ L;X) < �1:Ep�sh , apì thn Prìtash 6.3.1(b),pos1(0)-Ind(Q2 \ L;X) � pos1(0)-Ind(Q2; X) = �2:'Ara, pos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X) < �1(+)�2 = �:Ef> ìson (Q1 [Q2) \ L = (Q1 \ L) [ (Q2 \ L), apì thn upìjesh th epagwg  ,pos1(0)-Ind((Q1 [Q2) \ L;X) < � + 1:Sunep¸ ,pos1(0)-Ind(Q1 [Q2; X) � � + 1 = pos1(0)-Ind(Q1; X)(+)pos1(0)-Ind(Q2; X) + 1;apodeiknÔonta thn prìtash. �6.5.3 Parat rhsh. Apì thn Prìtash 6.2.12 kai to Pìrisma 6.2.13 prokÔptei ìti to para-p�nw Je¸rhma Ajro�smato e�nai tetrimmèno gia ti diast�sei p1(1)-Ind kai pos1(1)-Ind.6.6 Je¸rhma TaÔtish gia ti pos1-ind kai pos1(0)-IndSthn enìthta aut  d�noume sunj ke ¸ste oi diast�sei -sunart sei jèsew pos1-indkai pos1(0)-Ind na sump�ptoun.6.6.1 Orismì . 'Estw df mia apì ti diast�sei -sunart sei jèsew tou tÔpou Ind:pi(j)-Ind, posi(j)-Ind, ìpou i 2 f0; 1g kai j 2 f0; 1g. Lème ìti to arijm simo je¸rhmaajro�smato gia th df isqÔei s> èna q¸ro X, e�n gia k�je arijm simh oikogèneia Qi,i = 1; 2; : : : ; apì kleistoÔ upoq¸rou tou X me df(Qi; X) � �, i = 1; 2; : : : ; èqoumedf([1i=1Qi; X) � �.6.6.2 L mma. 'Estw X èna Lindel�of kanonikì q¸ro kai Q èna kleistì uposÔnolo touX. E�n pos1-ind(Q;X) � �, tìte gia k�je zeÔgo (A;B) apì xèna metaxÔ tou kleist�

118 Kef�laio 6uposÔnola tou X up�rqei èna uposÔnolo L tou X to opo�o diaqwr�zei ta A kai B ètsi¸ste Q \ L = Q \ [1i=1Li, ìpou to Li e�nai kleistì sto X kai pos1-ind(Q \ Li; X) < �,i = 1; 2; : : : :Apìdeixh. 'Estw (A;B) zeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X. Ef>ìson o q¸ro X e�nai fusikì , up�rqoun dÔo anoikt� uposÔnola U kai V tou X ètsi¸ste A � U , B � W kai ClX(U) \ ClX(W ) = ;. Ef> ìson o X e�nai Lindel�of kaipos1-ind(Q;X) � �, up�rqei mia arijm simh anoikt  eklèptunshfV1; V2; : : : ; X n (Q [ ClX(U)); X n (Q [ ClX(W ))gtou anoiktoÔ kalÔmmato fX n ClX(U); X nClX(W )g tou X tètoia ¸steClX(Vi) \ A = ;   ClX(Vi) \B = ;kai pos1-ind(Q \ BdX(Vi); X) < �; i = 1; 2; : : : :Apì to L mma 2.3.16 tou [19℄, up�rqei èna uposÔnolo L tou X to opo�o diaqwr�zei ta Akai B ètsi ¸ste Q \ L = Q \ [1i=1Li, ìpou Li = L \ BdX(Vi), i = 1; 2; : : : : Profan¸ ,pos1-ind(Q \ Li; X) � pos1-ind(Q \ BdX(Vi); X) < �; i = 1; 2; : : : : �6.6.3 Prìtash. 'Estw X èna Lindel�of kanonikì q¸ro ston opo�o to arijm simo je¸-rhma ajro�smato gia th pos1(0)-Ind isqÔei kai Q èna kleistì uposÔnolo tou X. Tìte,èqoume pos1-ind(Q;X) = pos1(0)-Ind(Q;X).Apìdeixh. Apì thn Prìtash 6.2.15(2), arke� na apode�xoume ìti(6:19) pos1(0)-Ind(Q;X) � pos1-ind(Q;X):'Estw pos1-ind(Q;X) = � 2 [f�1;1g. H sqèsh (6.19) e�nai profan  e�n � = �1  � = 1. Upojètoume ìti � 2 O kai ìti h sqèsh (6.19) e�nai alhj  gia k�je uposÔ-nolo QY enì Lindel�of kanonikoÔ q¸rou Y me pos1-ind(QY ; Y ) < �. ApodeiknÔoume ìtipos1(0)-Ind(Q;X) � �.Apì thn Prìtash 6.4.2(b), arke� na apode�xoume ìti gia k�je zeÔgo (A;B) apì xènametaxÔ tou kleist� uposÔnola tou X, ìpou A � Q, up�rqei èna uposÔnolo L tou Xto opo�o diaqwr�zei ta A kai B ètsi ¸ste pos1(0)-Ind(Q \ L;X) < �. 'Estw (A;B)zeÔgo apì xèna metaxÔ tou kleist� uposÔnola tou X. Apì to L mma 6.6.2, up�rqei èna

Diast�sei -sunart sei jèsew tou tÔpou Ind 119uposÔnolo L tou X to opo�o diaqwr�zei ta A kai B ètsi ¸ste L = [1i=1Li, ìpou to Lie�nai kleistì sto X kai pos1(0)-Ind(Q \ Li; X) < �, i = 1; 2; : : : : Ef> ìson to arijm simoje¸rhma ajro�smato gia thn pos1(0)-Ind isqÔei sto X,pos1(0)-Ind(Q \ L;X) = pos1(0)-Ind(Q \ [1i=1Li; X)= pos1(0)-Ind([1i=1(Q \ Li); X) < �:Sunep¸ , pos1(0)-Ind(Q;X) � �. �6.7 Jewr mata Ginomènou6.7.1 Orismì . 'Estw df mia apì ti diast�sei -sunart sei jèsew tou tÔpou Ind:pi(j)-Ind, posi(j)-Ind, ìpou i 2 f0; 1g kai j 2 f0; 1g. Lème ìti to peperasmèno je¸rhmaajro�smato gia th df isqÔei s> èna q¸roX, e�n gia k�je zeÔgo (Q1,Q2) apì kleistoÔ upoq¸rou tou X me df(Q1; X) � � kai df(Q2; X) � � èqoume df(Q1 [Q2; X) � �.6.7.2 Prìtash. 'Estw X kai Y dÔo sumpage� q¸roi kai QX , QY dÔo kleist� uposÔnolatwn X kai Y , ant�stoiqa.(1) E�n to peperasmèno je¸rhma ajro�smato gia thn p1(0)-Ind isqÔei sto q¸ro X � Y ,tìte p1(0)-Ind(QX �QY ; X � Y ) � p1(0)-Ind(QX ; X)(+)p1(0)-Ind(QY ; Y ).(2) E�n to peperasmèno je¸rhma ajro�smato gia thn pos1(0)-Ind isqÔei sto q¸ro X�Y ,tìte pos1(0)-Ind(QX �QY ; X � Y ) � pos1(0)-Ind(QX ; X)(+)pos1(0)-Ind(QY ; Y ).Apìdeixh. (1) ApodeiknÔoume th sqèsh(6:20) p1(0)-Ind(QX �QY ; X � Y ) � p1(0)-Ind(QX ; X)(+)p1(0)-Ind(QY ; Y )me epagwg . E�n p1(0)-Ind(QX ; X)(+)p1(0)-Ind(QY ; Y ) = �1, tìte ta QX kai QY e�naiken� kai epomènw p1(0)-Ind(QX � QY ; X � Y ) = �1. Upojètoume ìti h sqèsh (6.20)e�nai alhj  gia k�je zeÔgh (QX ; X) kai (QY ; Y ) mep1(0)-Ind(QX ; X)(+)p1(0)-Ind(QY ; Y ) < �;ìpou to � e�nai èna stajerì diataktikì arijmì . JewroÔme dÔo zeÔgh (QX ; X) kai(QY ; Y ) me p1(0)-Ind(QX ; X)(+)p1(0)-Ind(QY ; Y ) = �. Arke� na apode�xoume ìtip1(0)-Ind(QX �QY ; X � Y ) � �:E�n p1(0)-Ind(QX ; X) = �1   p1(0)-Ind(QY ; Y ) = �1, tìte QX � QY = ; kai sunep¸ p1(0)-Ind(QX �QY ; X � Y ) = �1 < �. 'Estwp1(0)-Ind(QX ; X) = �

120 Kef�laio 6kai p1(0)-Ind(QY ; Y ) = ;ìpou �; 2 O.JewroÔme èna zeÔgo (F; U) apì uposÔnola tou X � Y , ìpou to F e�nai kleistìuposÔnolo tou X �Y , to U e�nai anoiktì uposÔnolo tou X �Y kai F � (QX �QY )\U .E�nai arketì na or�soume èna anoiktì uposÔnolo V tou X � Y ètsi ¸ste F � V � U kaip1(0)-Ind((QX �QY ) \ BdX�Y (V ); X � Y ) < �:Apì th sumpag�a tou F , up�rqei peperasmèno pl jo apì anoikt� uposÔnola V X1 ; : : : ; V Xntou X kai anoikt� uposÔnola V Y1 ; : : : ; V Yn tou Y ètsi ¸steF � V = [ni=1(V Xi � V Yi ) � U ,p1(0)-Ind(QX \ BdX(V Xi ); X) < � kaip1(0)-Ind(QY \ BdY (V Yi ); Y ) < gia i = 1; : : : ; n.'Eqoume(QX �QY ) \ BdX�Y (V ) = (QX �QY ) \ BdX�Y ([ni=1(V Xi � V Yi ))� [ni=1((QX �QY ) \ BdX�Y (V Xi � V Yi ))� [ni=1((QX �QY ) \ ((X � BdY (V Yi )) [ (BdX(V Xi )� Y )))� [ni=1((QX � (QY \ BdY (V Yi ))) [ ((QX \ BdX(V Xi ))�QY )),p1(0)-Ind(QX ; X)(+)p1(0)-Ind(QY \ BdY (V Yi ); Y ) < �(+) = �kai p1(0)-Ind(QX \ BdX(V Xi ); X)(+)p1(0)-Ind(QY ; Y ) < �(+) = �:Apì thn upìjesh th epagwg  , èqoumep1(0)-Ind(QX � (QY \ BdY (V Yi )); X � Y ) < �kai p1(0)-Ind((QX \ BdX(V Xi ))�QY ; X � Y ) < �:Ef> ìson to peperasmèno je¸rhma ajro�smato gia thn p1(0)-Ind isqÔei,p1(0)-Ind((QX �QY ) \ BdX�Y (V ); X � Y ) < �:Sunep¸ , p1(0)-Ind(QX �QY ; X � Y ) � �.

Diast�sei -sunart sei jèsew tou tÔpou Ind 121(2) ApodeiknÔoume th sqèsh(6:21) pos1(0)-Ind(QX �QY ; X � Y ) � pos1(0)-Ind(QX ; X)(+)pos1(0)-Ind(QY ; Y )me epagwg . E�n pos1(0)-Ind(QX ; X)(+)pos1(0)-Ind(QY ; Y ) = �1, tìte ta QX kai QYe�nai ken� kai epomènw pos1(0)-Ind(QX � QY ; X � Y ) = �1. Upojètoume ìti h sqèsh(6.21) e�nai alhj  gia k�je zeÔgh (QX ; X) kai (QY ; Y ) mepos1(0)-Ind(QX ; X)(+)pos1(0)-Ind(QY ; Y ) < �;ìpou to � e�nai èna stajerì diataktikì arijmì . JewroÔme dÔo zeÔgh (QX ; X) kai(QY ; Y ) me pos1(0)-Ind(QX ; X)(+)pos1(0)-Ind(QY ; Y ) = �. Arke� na apode�xoume ìtipos1(0)-Ind(QX �QY ; X � Y ) � �:E�n pos1(0)-Ind(QX ; X) = �1   pos1(0)-Ind(QY ; Y ) = �1, tìteQX�QY = ; kai sunep¸ pos1(0)-Ind(QX �QY ; X � Y ) = �1 < �. 'Estwpos1(0)-Ind(QX ; X) = �kai pos1(0)-Ind(QY ; Y ) = ;ìpou �; 2 O.JewroÔme èna zeÔgo (F; U) apì uposÔnola tou X � Y , ìpou to F e�nai kleistìuposÔnolo tou X �Y , to U e�nai anoiktì uposÔnolo tou X �Y kai F � (QX �QY )\U .E�nai arketì na or�soume èna anoiktì uposÔnolo V tou X � Y ètsi ¸ste F � V � U kaipos1(0)-Ind((QX �QY ) \ BdX�Y (V ); X � Y ) < �:Apì th sumpag�a tou F , up�rqei peperasmèno pl jo apì anoikt� uposÔnola V X1 ; : : : ; V Xntou X kai anoikt� uposÔnola V Y1 ; : : : ; V Yn tou Y ètsi ¸steF � V = [ni=1(V Xi � V Yi ) � U ,pos1(0)-Ind(QX \ BdX(V Xi ); X) < � kaipos1(0)-Ind(QY \ BdY (V Yi ); Y ) < gia i = 1; : : : ; n.'Eqoume(QX �QY ) \ BdX�Y (V ) = (QX �QY ) \ BdX�Y ([ni=1(V Xi � V Yi ))� [ni=1((QX �QY ) \ BdX�Y (V Xi � V Yi ))

122 Kef�laio 6� [ni=1((QX �QY ) \ ((X � BdY (V Yi )) [ (BdX(V Xi )� Y )))� [ni=1((QX � (QY \ BdY (V Yi ))) [ ((QX \ BdX(V Xi ))�QY )),pos1(0)-Ind(QX ; X)(+)pos1(0)-Ind(QY \ BdY (V Yi ); Y ) < �(+) = �kai pos1(0)-Ind(QX \ BdX(V Xi ); X)(+)pos1(0)-Ind(QY ; Y ) < �(+) = �:Apì thn upìjesh th epagwg  , èqoumepos1(0)-Ind(QX � (QY \ BdY (V Yi )); X � Y ) < �kai pos1(0)-Ind((QX \ BdX(V Xi ))�QY ; X � Y ) < �:Ef> ìson to peperasmèno je¸rhma ajro�smato gia thn pos1(0)-Ind isqÔei,pos1(0)-Ind((QX �QY ) \ BdX�Y (V ); X � Y ) < �:Sunep¸ , pos1(0)-Ind(QX �QY ; X � Y ) � �. �

Kef�laio 7Diast�sei -sunart sei b�sew jèsew tou tÔpou dimSto bibl�o [37℄ or�sjhkan diast�sei -sunart sei b�sew tou tÔpou ind, Ind kai dim.Oi diast�sei -sunart sei autè melet jhkan mìno w pro thn idiìthta th kajolikìth-ta . Sto kef�laio autì d�nontai diast�sei -sunart sei b�sew jèsew tou tÔpou dim kaiapodeiknÔetai h idiìthta th kajolikìthta gia ti sunart sei autè . Ta apotelèsmatatou kefala�ou autoÔ e�nai ìla prwtìtupa.Se ì,ti akolouje� me th lèxh q¸ro ja ennooÔme ènan T0-q¸ro me b�ro � � .7.1 Basiko� orismo�Sthn ergas�a [71℄ or�sjhkan kai melet jhkan dÔo diast�sei jèsew dim kai dim�.Parak�tw oi diast�sei dim kai dim� sumbol�zontai p0-dim kai p1-dim, ant�stoiqa.7.1.1 Orismì . JewroÔme th sun�rthsh p0-dim me ped�o orismoÔ thn kl�sh ìlwn twnzeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n to sÔnolo! [ f�1;1g pou ikanopoie� thn parak�tw sunj kh:p0-dim(Q;X) � n; ìpou n 2 f�1g [ !e�n kai mìnon e�n gia k�je peperasmèno anoiktì k�lumma tou q¸rou X up�rqei ènapeperasmèno anoiktì k�lumma r tou Q, eklèptunsh tou , ètsi ¸ste ord(r) � n.7.1.2 Orismì . JewroÔme th sun�rthsh p1-dim me ped�o orismoÔ thn kl�sh ìlwn twnzeug¸n (Q;X), ìpou Q e�nai èna uposÔnolo enì q¸rou X, kai ped�o tim¸n to sÔnolo! [ f�1;1g pou ikanopoie� thn parak�tw sunj kh:p1-dim(Q;X) � n; ìpou n 2 f�1g [ !123

124 Kef�laio 7e�n kai mìnon e�n gia k�je peperasmèno anoiktì k�lumma tou q¸rou X up�rqei miapeperasmènh oikogèneia r apì anoikt� uposÔnola tou X, eklèptunsh tou , ètsi ¸steQ � [fV : V 2 rg kai ord(r) � n.7.1.3 Parat rhsh. ParathroÔme ìti gia k�je uposÔnolo Q enì q¸rou X èqoumep0-dim(Q;X) � p1-dim(Q;X) � dim(X):Ep�sh , e�n X e�nai to ep�pedo tou Niemytzki kai Q = f(x; 0) : x 2 Rg, tìtep0-dim(Q;X) < p1-dim(Q;X) < dim(X)kai dim(Q) 6= p1-dim(Q;X):(Blèpe Example 5 tou [71℄).7.1.4 Orismì . (1) 'Estw B mia b�sh enì q¸rou X. 'Ena k�lumma tou X lègetaiB-k�lumma, e�n ìla ta stoiqe�a tou e�nai stoiqe�a th B.(2) 'Estw Q èna uposÔnolo enì q¸rou X. Lème ìti mia oikogèneia apì uposÔnola touX kalÔptei to Q, e�n h oikogèneia fQ \ U : U 2 g e�nai èna k�lumma tou upoq¸rou Q.7.1.5 Orismì . 'Estw IF mia kl�sh uposunìlwn kai Q èna uposÔnolo tou X. 'Enak�lumma tou X (ant�stoiqa, mia oikogèneia apì uposÔnola tou X pou kalÔptei toQ) lègetai IF-k�lumma (ant�stoiqa, IF-k�lumma gia to Q), e�n (C;X) 2 IF gia k�jeC 2 . 'Ena IF-k�lumma (ant�stoiqa, èna IF-k�lumma gia to Q) r, pou e�nai eklèptunsh enì kalÔmmato tou X (ant�stoiqa, mia oikogèneia apì uposÔnola tou X pou kalÔptounto Q), kale�tai IF-eklèptunsh tou (ant�stoiqa, IF-eklèptunsh tou gia to Q).7.1.6 Sumbolismì . H kl�sh ìlwn twn p-b�sewn (ant�stoiqa, ìlwn twn pos-b�sewn  ps-b�sewn), dhlad  h kl�sh ìlwn twn tri�dwn (Q;B;X), ìpou Q e�nai èna uposÔnoloenì q¸rou X kai B e�nai mia p-b�sh (ant�stoiqa, mia pos-b�sh   ps-b�sh) gia to Q stoX, sumbol�zetai me ID(p-base) (ant�stoiqa, me ID(pos-base)   ID(ps-base)).7.1.7 Orismì . Gia k�je kl�sh IF uposunìlwn jewroÔme th sun�rthsh b-p0-dimIF meped�o orismoÔ thn kl�sh ID(ps-base) kai ped�o tim¸n to sÔnolo ![f�1;1g pou ikanopoie�thn parak�tw sunj kh:b-p0-dimIF(Q;B;X) � n; ìpou n 2 f�1g [ !e�n kai mìnon e�n gia k�je peperasmèno B-k�lumma tou q¸rou X up�rqei èna pepera-smèno IF-k�lumma r tou Q, eklèptunsh tou , ètsi ¸ste ord(r) � n.

Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 125E�n IF e�nai h kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna anoiktì uposÔnoloenì q¸rou X, tìte h sun�rthsh b-p0-dimIF ja sumbol�zetai me b-p0-dimOp.7.1.8 Orismì . Gia k�je kl�sh IF uposunìlwn jewroÔme th sun�rthsh b-p1-dimIF meped�o orismoÔ thn kl�sh ID(ps-base) kai ped�o tim¸n to sÔnolo ![f�1;1g pou ikanopoie�thn parak�tw sunj kh:b-p1-dimIF(Q;B;X) � n; ìpou n 2 f�1g [ !e�n kai mìnon e�n gia k�je peperasmènoB-k�lumma tou q¸rouX up�rqei mia peperasmènhoikogèneia r apì anoikt� uposÔnola tou X, eklèptunsh tou , ètsi ¸ste (C;X) 2 IF giak�je C 2 , Q � [fV : V 2 rg kai ord(r) � n.E�n IF e�nai h kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna anoiktì uposÔnoloenì q¸rou X, tìte h sun�rthsh b-p1-dimIF ja sumbol�zetai me b-p1-dimOp.7.1.9 Prìtash. 'Estw B mia b�sh enì q¸rou X. Gia k�je uposÔnolo Q tou X èqoume(1) b-p0-dimOp(Q;B;X) � p0-dim(Q;X) kai(2) b-p1-dimOp(Q;B;X) � p1-dim(Q;X).Apìdeixh. (1) ApodeiknÔoume ìti(7:1) b-p0-dimOp(Q;B;X) � p0-dim(Q;X):'Estw p0-dim(Q;X) = n 2 ! [ f�1;1g. H anisìthta (7.1) e�nai profan  e�n n = �1  n = 1. Upojètoume ìti n 2 !. 'Estw èna peperasmèno B-k�lumma tou q¸rou X.Profan¸ , h oikogèneia e�nai èna peperasmèno anoiktì k�lumma tou q¸rou X. Ef> ìsonp0-dim(Q;X) = n, up�rqei èna peperasmèno anoiktì k�lumma r tou Q, eklèptunsh tou ,ètsi ¸ste ord(r) � n. Sunep¸ , b-p0-dimOp(Q;B;X) � n.(2) ApodeiknÔoume ìti(7:2) b-p1-dimOp(Q;B;X) � p1-dim(Q;X):'Estw p1-dim(Q;X) = n 2 ! [ f�1;1g. H anisìthta (7.2) e�nai profan  e�n n = �1  n = 1. Upojètoume ìti n 2 !. 'Estw èna peperasmèno B-k�lumma tou q¸rou X.Profan¸ , h oikogèneia e�nai èna peperasmèno anoiktì k�lumma tou q¸rou X. Ef>ìson p1-dim(Q;X) = n, up�rqei mia peperasmènh oikogèneia r apì anoikt� uposÔnolatou X, eklèptunsh tou , ètsi ¸ste Q � [fV : V 2 rg kai ord(r) � n. Sunep¸ ,b-p1-dimOp(Q;B;X) � n. �

126 Kef�laio 77.1.10 Par�deigma. 'Estw R to sÔnolo twn pragmatik¸n arijm¸n me th sun jh topo-log�a, Q = (�1; 2℄ kai B = f(a; b) : a; b 2 R kai a < bg [ fRg. Profan¸ h oikogèneiaB e�nai mia b�sh gia to R. ParathroÔme ìtip0-dim(Q;R) = p1-dim(Q;R) = 1kai b-p0-dimOp(Q;B;R) = b-p1-dimOp(Q;B;R) = 0:7.1.11 Orismì . Lème ìti mia kl�sh IF uposunìlwn e�nai kleist  w pro ton upì-qwro Q tou q¸rou X, e�n (Y \Q;Q) 2 IF gia k�je (Y;X) 2 IF.7.1.12 Prìtash. 'Estw B mia b�sh enì q¸rou X. Gia k�je uposÔnolo Q tou X èqoume(7:3) b-p0-dimIF(Q;B;X) � b-p1-dimIF(Q;B;X);ìpou IF e�nai kl�sh uposunìlwn kleist  w pro ton upìqwro Q tou q¸rou X.Apìdeixh. 'Estw b-p1-dimIF(Q;B;X) = n 2 ! [ f�1;1g. H anisìthta (7.3) e�naiprofan  e�n n = �1   n = 1. Upojètoume ìti n 2 !. 'Estw èna peperasmènoB-k�lumma tou q¸rou X. Ef> ìson b-p1-dimIF(Q;B;X) = n, up�rqei mia peperasmènhoikogèneia r apì uposÔnola tou X, eklèptunsh tou , ètsi ¸ste (C;X) 2 IF gia k�jeC 2 , Q � [fV : V 2 rg kai ord(r) � n. Tìte, h oikogèneia rQ = fU \ Q : U 2 rge�nai èna peperasmèno IF-k�lumma tou Q, eklèptunsh tou , kai èqei t�xh � n. Sunep¸ ,b-p0-dimIF(Q;B;X) � n. �7.1.13 Par�deigma. 'Estw X to ep�pedo tou Niemytzki kai Q = f(x; 0) : x 2 Rg. E�nsumbol�soume me t thn topolog�a tou X, tìteb-p0-dimOp(Q; t;X) = p0-dim(Q;X) < p1-dim(Q;X) = b-p1-dimOp(Q; t;X):(Blèpe Example 5 tou [71℄).7.1.14 Orismì . (Blèpe [37℄) Gia k�je kl�sh IF uposunìlwn jewroÔme th sun�rthshb-dimIF me ped�o orismoÔ thn kl�sh ìlwn twn zeug¸n (B;X), ìpou B e�nai mia b�sh enì q¸rou X, kai ped�o tim¸n to sÔnolo ! [ f�1;1g pou ikanopoie� thn parak�tw sunj kh:b-dimIF(B;X) � n; ìpou n 2 f�1g [ !e�n kai mìnon e�n gia k�je peperasmènoB-k�lumma tou q¸rouX up�rqei mia peperasmènhIF-eklèptunsh r tou me ord(r) � n.

Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 127E�n IF e�nai h kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna anoiktì uposÔnoloenì q¸rou X, tìte h sun�rthsh b-dimIF ja sumbol�zetai me b-dimOp.7.1.15 Prìtash. Gia k�je q¸ro X kai gia k�je b�sh B tou X èqoume(7:4) b-dimOp(B;X) � dim(X):Epiplèon, e�n h b�sh B e�nai peperasmènh, tìteb-dimOp(B;X) = dim(X):Apìdeixh. 'Estw dim(X) = n 2 ! [ f�1;1g. H anisìthta (7.4) e�nai profan  e�nn = �1   n = 1. Upojètoume ìti n 2 !. 'Estw èna peperasmèno B-k�lumma touq¸rou X. Profan¸ , h oikogèneia e�nai èna peperasmèno anoiktì k�lumma tou q¸rouX. Ef> ìson dim(X) = n, h oikogèneia èqei mia peperasmènh eklèptunsh r apì anoikt�uposÔnola tou X me ord(r) � n. Sunep¸ , b-dimOp(B;X) � n.'Estw ìti h b�sh B e�nai peperasmènh. ApodeiknÔoume ìti b-dimOp(B;X) = dim(X).Apì th sqèsh (7.4), arke� na apode�xoume ìti(7:5) dim(X) � b-dimOp(B;X):'Estw b-dimOp(B;X) = n 2 ! [ f�1;1g. H anisìthta (7.5) e�nai profan  e�n n = �1  n =1. Upojètoume ìti n 2 !. 'Estw = fUi : i = 1; : : : ; kg èna peperasmèno anoiktìk�lumma tou q¸rou X. Ef> ìson h b�sh B e�nai peperasmènh, gia k�je i = 1; : : : ; kup�rqoun stoiqe�aWi; j, j = 1; : : : ; ki th b�sh B ètsi ¸ste Ui = Skij=1Wi; j. H oikogèneia 0 = fWi; j : j = 1; : : : ; ki kai i = 1; : : : ; kge�nai èna peperasmèno B-k�lumma tou q¸rou X. Ef> ìson b-dimOp(B;X) = n, up�rqei miapeperasmènh eklèptunsh r tou 0 apì anoikt� uposÔnola tou X me ord(r) � n. Profan¸ ,h oikogèneia r e�nai mia peperasmènh eklèptunsh tou . Sunep¸ , dim(X) � n. �7.1.16 Prìtash. 'Estw IF h kl�sh ìlwn twn uposunìlwn kai B mia b�sh enì q¸rouX. Tìte, b-dimIF(B;X) = 0.Apìdeixh. 'Estw = fUi : i = 1; : : : ; kg èna peperasmèno B-k�lumma tou q¸rou X. TasÔnola V1 = U1 kai Vi = Ui n[j<iUj; ìpou i = 2; : : : ; k;

128 Kef�laio 7apoteloÔn mia peperasmènh IF-eklèptunsh tou me t�xh 0. 'Ara, b-dimIF(B;X) = 0. �7.1.17 Orismì . Gia k�je kl�sh IF uposunìlwn jewroÔme th sun�rthsh b-p-dimIF meped�o orismoÔ thn kl�sh ID(p-base) kai ped�o tim¸n to sÔnolo ![f�1;1g pou ikanopoie�thn parak�tw sunj kh:b-p-dimIF(Q;B;X) � n; ìpou n 2 f�1g [ !e�n kai mìnon e�n gia k�je peperasmènh oikogèneia apì stoiqe�a th B pou kalÔptei toQ up�rqei mia peperasmènh IF-eklèptunsh r tou gia to Q me ord(r) � n.E�n IF e�nai h kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna anoiktì uposÔnoloenì q¸rou X, tìte h sun�rthsh b-p-dimIF ja sumbol�zetai me b-p-dimOp.7.1.18 Parat rhsh. (1) E�n ston Orismì 7.1.17 antikatast soume thn kl�sh ID(p-base) me thn kl�sh ID(pos-base) (ant�stoiqa, me thn kl�sh ID(ps-base)), tìte ja p�roumeth di�stash-sun�rthsh b�sew jèsew b-pos-dimIF (ant�stoiqa, b-ps-dimIF). Profan¸ ,(a) b-p-dimIFjID(pos-base) = b-pos-dimIF kai(b) b-p-dimIFjID(ps-base) = b-ps-dimIF.(2) 'Estw IF mia kl�sh uposunìlwn kai B mia b�sh enì q¸rou X. Tìte,b-ps-dimIF(X;B;X) = b-p0-dimIF(X;B;X)= b-p1-dimIF(X;B;X)= b-dimIF(B;X):7.1.19 Prìtash. 'Estw B mia b�sh enì q¸rou X. Gia k�je uposÔnolo Q tou X èqoume(7:6) b-p1-dimIF(Q;B;X) � b-ps-dimIF(Q;B;X):Apìdeixh. 'Estw b-ps-dimIF(Q;B;X) = n 2 ![f�1;1g. H anisìthta (7.6) e�nai profa-n  e�n n = �1   n =1. Upojètoume ìti n 2 !. 'Estw èna peperasmèno B-k�lumma touq¸rou X. Tìte, h oikogèneia kalÔptei to Q. Ef> ìson b-ps-dimIF(Q;B;X) = n, up�rqeimia peperasmènh IF-eklèptunsh r tou me ord(r) � n. Profan¸ , Q � [fV : V 2 rg.Sunep¸ , b-p1-dimIF(Q;B;X) � n. �7.1.20 Prìtash. 'Estw IF mia kl�sh uposunìlwn kai B mia b�sh enì q¸rou X. Giak�je uposÔnolo Q tou X èqoume b-p1-dimIF(Q;B;X) � b-dimIF(B;X). Ep�sh , gia k�jeV 2 B èqoume b-ps-dimIF(X n V;B;X) � b-dimIF(B;X).

Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 129Apìdeixh. ApodeiknÔoume ìti(7:7) b-p1-dimIF(Q;B;X) � b-dimIF(B;X):'Estw b-dimIF(B;X) = n 2 ! [ f�1;1g. H anisìthta (7.7) e�nai profan  e�n n = �1  n = 1. Upojètoume ìti n 2 !. 'Estw èna peperasmèno B-k�lumma tou q¸rou X. Ef>ìson b-dimIF(B;X) = n, up�rqei mia peperasmènh IF-eklèptunsh r tou me ord(r) � n.Profan¸ , Q � [fV : V 2 rg. Sunep¸ , b-p1-dimIF(Q;B;X) � n.ApodeiknÔoume ìti(7:8) b-ps-dimIF(X n V;B;X) � b-dimIF(B;X):'Estw b-dimIF(B;X) = n 2 ! [ f�1;1g. H anisìthta (7.8) e�nai profan  e�n n = �1  n =1. Upojètoume ìti n 2 !. 'Estw mia peperasmènh oikogèneia apì stoiqe�a th B poukalÔptei to X nV . Tìte, h oikogèneia [fV g e�nai èna peperasmèno B-k�lumma tou q¸rouX. Ef> ìson b-dimIF(B;X) = n, up�rqei mia peperasmènh IF-eklèptunsh r tou [fV g meord(r) � n. Profan¸ , X n V � [fV : V 2 rg. Sunep¸ , b-ps-dimIF(X n V;B;X) � n.�7.1.21 Pìrisma. 'Estw IF mia kl�sh uposunìlwn kai B mia b�sh enì q¸rou X. Tìte,b-dimIF(B;X) = maxfb-ps-dimIF(X n V;B;X) : V 2 Bg= maxfb-p1-dimIF(Q;B;X) : Q � Xg:Apìdeixh. Apì thn Prìtash 7.1.20, èqoumesupfb-p1-dimIF(Q;B;X) : Q � Xg � b-dimIF(B;X)kai supfb-ps-dimIF(X n V;B;X) : V 2 Bg � b-dimIF(B;X):Ep�sh , gia Q = X èqoumeb-p1-dimIF(Q;B;X) = b-dimIF(B;X)kai gia V = ; èqoume b-ps-dimIF(X;B;X) = b-dimIF(B;X):'Ara,

130 Kef�laio 7b-dimIF(B;X) = maxfb-p1-dimIF(Q;B;X) : Q � Xg kaib-dimIF(B;X) = maxfb-ps-dimIF(X n V;B;X) : V 2 Bg. �7.1.22 Par�deigma. 'Estw X = fa; b; g. JewroÔme ep� tou X thn topolog�at = f;; fa; bg; fb; g; fbg; Xg:'Estw Q = f g. ParathroÔme ìtib-p0-dimOp(Q; t;X) = b-p1-dimOp(Q; t;X) = 0 kaib-dimOp(t; X) = 1.7.1.23 Par�deigma. 'Estw R to sÔnolo twn pragmatik¸n arijm¸n me th sun jh topolo-g�a, I = [0; 1℄ kai B = f(a; b) : a; b 2 R kai a < bg [ fRg. Profan¸ h oikogèneia B e�naimia b�sh gia to R kai dim(R) = 1. Epiplèon, epeid  to monosÔnolo fRg e�nai eklèptunshk�je peperasmènou B-kalÔmmato tou R, b-dimOp(B;R) = 0. ParathroÔme ìtib-p0-dimOp(I; B;R) = b-p1-dimOp(I; B;R) = 0 kaib-ps-dimOp(I; B;R) = 1.7.1.24 Parat rhsh. Oi sqèsei metaxÔ twn diast�sewn-sunart sewn tou tÔpou dim(blèpe ti parap�nw prot�sei kai parade�gmata) sunoy�zontai sto parak�tw di�gramma,ìpou <<!>> shma�nei <<� >> kai <<9>> shma�nei ìti << genik� � >>.b-dimOp(B;X)���

// dim(X)���

�oob-ps-dimOp(Q;B;X) � // b-p1-dimOp(Q;B;X)���

OO

//oo p1-dim(Q;X)�oo ���

OO

b-p0-dimOp(Q;B;X) //

OO p0-dim(Q;X)�oo

OO

Di�gramma 7.17.2 Kajolik� stoiqe�a gia diast�sei -sunart sei b�-sew jèsew tou tÔpou dim7.2.1 Sumbolismo�. 'Estw df mia apì ti parak�tw diast�sei -sunart sei b�sew jè-sew tou tÔpou dim: b-p-dimIF, b-pos-dimIF, b-ps-dimIF, b-p0-dimIF kai b-p1-dimIF. Gia

Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 131k�je n 2 f�1g [ ! me IP(df � n) sumbol�zoume thn kl�sh p-b�sewn pou apotele�tai apììle ti tri�de (Q;B;X) me df(Q;B;X) � n.7.2.2 Orismì . (Blèpe [37℄) 'Estw X èna q¸ro . Lègetai ìti mia kl�sh IF uposunìlwnikanopoie� th Sunj kh Peperasmènh 'Enwsh (Finite Union Condition), e�n apì ti sunj ke (Fi; X) 2 IF, i 2 j 2 !, prokÔptei ìti ([fFi : i 2 jg; X) 2 IF. Ep�sh , lègetai ìtimia kl�sh IF uposunìlwn ikanopoie� th Sunj kh KenoÔ Uposunìlou (Empty SubsetCondition), e�n apì th sunj kh (Q;X) 2 IF prokÔptei ìti (;; X) 2 IF.7.2.3 L mma. (Blèpe [37℄) 'Estw Q èna periorismì mia diktuwmènh oikogèneia S apìq¸rou . Upojètoume ìti o Q e�nai (M0;R0)-pl rh periorismì gia k�poio sun-shm�diM0 th S kai gia k�poia (M0;Q)-epitrept  oikogèneia apì sqèsei isodunam�a ep� th S.Tìte, gia k�je sun-shm�diM th S, pou e�nai sun-epèktash tou M0 kai gia k�je (M;Q)-epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, pou e�nai telik¸ leptìterh th R0, o Q e�nai (M;R)-pl rh periorismì .7.2.4 Je¸rhma. 'Estw df mia apì ti diast�sei -sunart sei b�sew jèsew : b-p-dimIF,b-pos-dimIF, b-ps-dimIF, b-p0-dimIF kai b-p1-dimIF. E�n h kl�sh IF e�nai koresmènh, pl rh kai ikanopoie� ti Sunj ke Peperasmènh 'Enwsh kai KenoÔ Uposunìlou, tìte gia k�jen 2 f�1g [ ! h kl�sh IP(df � n) e�nai koresmènh.Apìdeixh. D�netai h apìdeixh tou jewr mato mìno gia thn kl�sh IP(b-p-dimIF � n),ìpou n 2 f�1g [ !. H apìdeixh tou jewr mato gia ti upìloipe kl�sei e�nai an�logh.'Estw n 2 f�1g [ !. Ja apode�xoume ìti h kl�sh IP(b-p-dimIF � n) e�nai koresmènh.'Estw S mia diktuwmènh oikogèneia apì q¸rou , Q � fQX : X 2 Sg èna periorismì th S, B � fBX : X 2 Sg mia IP(b-p-dimIF � n)-sun-p-b�sh gia ton Q sthn S kaiN � ffV X" : " 2 �g : X 2 Sgmia sun-diktÔwsh th B. ApodeiknÔoume ìti up�rqei mia sun-epèktashM+ touN ètsi ¸stegia k�je sun-shm�di M th S, pou e�nai sun-epèktash tou M+, na up�rqei mia (M;Q)-epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S tètoia ¸ste gia k�je epitrept oikogèneia R apì sqèsei isodunam�a ep� th S, pou e�nai telik¸ leptìterh th R+, kaik�je L;H;E 2 C}(R) me L � H � E na èqoume ìti(T(EjQ);BH};�(�);T(L)) 2 IP(b-p-dimIF � n);ìpou � e�nai mia endeiktik  sun�rthsh apì to N sto M.

132 Kef�laio 7Pr¸ta, gia k�je q 2 F n f;g kai gia k�je " 2 q ja kataskeu�soume èna IF-periorismìth S: W(q; ") � fWX(q; ") : X 2 Sg:'Estw q = f"0; : : : ; "kg èna stoiqe�o tou Fnf;g. Gia k�je X 2 S jewroÔme èna diktuwmènosÔnolo V X(q) � fV X" : " 2 qg:E�n to sÔnolo V X(q) den kalÔptei to QX , tìte gia k�je " 2 q jètoume WX(q; ") = ;.Upojètoume ìti to sÔnolo V X(q) kalÔptei to QX . Ef> ìson h V X(q) e�nai mia peperasmènhoikogèneia apì stoiqe�a th BX pou kalÔptei to QX kai (QX ; BX ; X) 2 IP(b-p-dimIF � n),up�rqei mia peperasmènh IF-eklèptunsh rXq tou V X(q) gia to QX me ord(rXq ) � n. JètoumeWX(q; "0) = [fV 2 rXq : V � V X"0 gkai WX(q; "i) = [fV 2 rXq : V � V X"i kai V * V X" gia k�je " 2 f"0; : : : ; "i�1gggia k�je i 2 f1; : : : ; kg. Ef> ìson to rXq e�nai èna IF-k�lumma gia to QX me ord(rXq ) � nkai epeid  h IF ikanopoie� ti Sunj ke Peperasmènh 'Enwsh kai KenoÔ Uposunìlou, tosÔnolo WX(q) � fWX(q; ") : " 2 qge�nai ep�sh èna IF-k�lumma gia to QX me ord(WX(q)) � n. Epiplèon, apì thn kataskeu ,WX(q; ") � V X" gia k�je " 2 q:Shmei¸noume ìti to k�lummaWX(q) èqei ep�sh thn parak�tw idiìthta: e�n ta "0; : : : ; "n+1e�nai diakekrimèna stoiqe�a tou q ètsi ¸ste WX(q; "i) 6= ;, i 2 f0; : : : ; n + 1g, tìte taWX(q; "0); : : : ;WX(q; "n+1) e�nai diakekrimèna stoiqe�a tou WX(q) kai epomènw WX(q; "0) \ : : : \WX(q; "n+1) = ;:'Estw M+ èna sun-shm�di th S, to opo�o e�nai sun-epèktash tou N. Sumbol�zoumeme �N mia endeiktik  sun�rthsh apì to N sto M+. Qwr� periorismì th genikìthta upojètoume ìti gia k�je q 2 F n f;g kai gia k�je " 2 q, to M+ e�nai èna arqikì sun-shm�di th S pou antistoiqe� ston periorismì W(q; ") kai sthn kl�sh IF. Epiplèon, ef>ìson h kl�sh IF e�nai pl rh mporoÔme na upojèsoume ìti up�rqei mia oikogèneia R+0 apìsqèsei isodunam�a ep� th S ètsi ¸ste gia k�je q 2 F n f;g kai gia k�je " 2 q naèqoume: (a) h oikogèneia R+0 e�nai (M+;W(q; "))-epitrept  kai (b) o periorismì W(q; ")

Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 133th S e�nai (M+;R+0 )-pl rh . Ja apode�xoume ìti to M+ e�nai èna arqikì sun-shm�dith S pou antistoiqe� ston periorismì Q, sth sun-diktÔwsh N th B kai sthn kl�shIP(b-p-dimIF � n). Pr�gmati, èstwM � ffUXÆ : Æ 2 �g : X 2 Sgmia auja�reth sun-epèktash tou M+. Sumbol�zoume me �+ mia endeiktik  sun�rthsh apìto M+ sto M. Tìte, to sun-shm�di M e�nai sun-epèktash tou sun-shmadioÔ N kai hsun�rthsh � = �+ Æ �N e�nai mia endeiktik  sun�rthsh apì to N sto M. (Sunep¸ ,V X" = UX�(") gia k�je " 2 � kai X 2 S).Gia k�je q 2 F n f;g kai gia k�je " 2 q jewroÔme mia arqik  oikogèneia R+q;" apìsqèsei isodunam�a ep� th S pou antistoiqe� sto sun-shm�diM, ston periorismì W(q; ")kai sthn kl�sh IF. Apì thn Prìtash 3.1.5(1), up�rqei mia epitrept  oikogèneia R+ apìsqèsei isodunam�a ep� th S, h opo�a e�nai telik¸ leptìterh apì thn R+0 kai apì ìle ti oikogèneie R+q;". Ja apode�xoume ìti h R+ e�nai mia arqik  oikogèneia th S pou anti-stoiqe� sto sun-shm�diM, ston periorismì Q, sth sun-diktÔwsh N th B kai sthn kl�shIP(b-p-dimIF � n). Pro toÔto jewroÔme mia auja�reth epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, h opo�a e�nai telik¸ leptìterh th R+. ApodeiknÔoume ìti gia k�jeL;H;E 2 C}(R) me E � H � L, èqoume (T(EjQ);BH};�(�);T(L)) 2 IP(b-p-dimIF � n).'Estw L;H;E stoiqe�a tou C}(R) me E � H � L kai � fUTÆ0(H0); : : : ; UTÆk(Hk)gmia peperasmènh oikogèneia apì stoiqe�a th BH};�(�), ìpou H0 � H; : : : ;Hk � H kaiÆ0; : : : ; Æk 2 �(�), pou kalÔptei to T(EjQ). Jètoume s = fÆ0; : : : ; Ækg kai q = ��1(s). Giak�je X 2 E sumbol�zoume me qX to sÔnolo ìlwn twn stoiqe�wn " tou q gia ta opo�aup�rqei i 2 f0; : : : ; kg ètsi ¸ste �(") = Æi kai X 2 Hi. Tìte, qX 6= ; kai to sÔnoloV X(qX) = fV X" : " 2 qXge�nai mia peperasmènh oikogèneia apì stoiqe�a tou BX pou kalÔptei to QX . Sunep¸ , toWX(qX) = fWX(qX ; ") : " 2 qXge�nai èna IF-k�lumma tou QX . 'Estw t èna stoiqe�o tou F tètoio ¸ste e�n K e�nai mia�t-kl�sh isodunam�a kai K\Hi 6= ; gia k�poio i 2 f0; : : : ; kg, tìte K � Hi. Profan¸ ,e�n X �t Y , tìte qX = qY .'Estw K èna stoiqe�o tou C(�t) tètoio ¸ste K � E. Jètoume qK = qX , ìpou X 2 K.Apì ta parap�nw, to qK den exart�tai apì to stoiqe�o X touK. JewroÔme thn oikogèneia (K) � fUTÆi (Hi) \ T(K) = UTÆi (K) : Æi 2 �(qK);K � Hig:

134 Kef�laio 7Profan¸ h oikogèneia (K) kalÔptei to T(KjQ). ApodeiknÔoume ìti hr(K) � fTjW(qK;") \ T(K) = T(KjW(qK;")) : " 2 qKge�nai mia IF-eklèptunsh tou (K) gia to T(KjQ) me ord(r(K)) � n. Ef> ìson h kl�sh IFe�nai koresmènh kl�sh uposunìlwn, to zeÔgo (T(KjW(qK;"));T(L))an kei sthn IF, dhlad  to r(K) e�nai èna IF-k�lumma gia to T(KjQ). Gia k�je X 2 K kaigia k�je " 2 qK èqoume WX(qK; ") � V X"kai sunep¸ eXT (WX(qK; ")) � eXT (V X" );ìpou eXT e�nai h fusik  emfÔteush tou X sto T. Apì thn Prìtash 2.4.26, èqoumeT(KjW(qK;")) = [feXT (WX(qK; ")) : X 2 Kg� feXT (V X" ) : X 2 Kg= feXT (UXÆi ) : X 2 Kg = UTÆi (K);ìpou i 2 f0; : : : ; kg ètsi ¸ste Æi = �(") kai K � Hi, pou shma�nei ìti h r(K) e�nai miaIF-eklèptunsh tou (K) gia to T(KjQ).T¸ra, ja apode�xoume ìti ord(r(K)) � n. Upojètoume ìti ord(r(K)) > n. Tìte,up�rqoun n+ 2 diakekrimèna stoiqe�a "0; : : : ; "n+1 tou qK ètsi ¸steT(KjW(qK;"0)) \ : : : \ T(KjW(qK;"n+1)) 6= ;  (TjW(qK;"0) \ T(K)) \ : : : \ (TjW(qK;"n+1) \ T(K)) 6= ;:'Estw a 2 TjW(qK;"0) \ : : : \ TjW(qK;"n+1) \ T(K)kai èstw (x;X) 2 a. Epeid  oi periorismo� W(qK; "i), i 2 f0; : : : ; n + 1g e�nai (M+;R+0 )-pl rei , apì to L mma 7.2.3, auto� oi periorismo� e�nai ep�sh (M;R)-pl rei . Sunep¸ ,x 2 WX(qX ; "0) \ : : : \WX(qX ; "n+1):Ef> ìson ta WX(q; "0); : : : ;WX(q; "n+1) e�nai diakekrimèna stoiqe�a tou WX(q), ta para-p�nw èrqontai se ant�fash me to gegonì ìti ord(WX(q)) � n. Sunep¸ , ord(r(K)) � n.

Diast�sei -sunart sei b�sew jèsew tou tÔpou dim 135Tèlo , parathroÔme ìti tor � [fr(K) : K 2 C(�t) kai K � Ege�nai mia IF-eklèptunsh tou gia to T(EjQ) me ord(r) � n. �7.2.5 Pìrisma. 'Estw df mia apì ti diast�sei -sunart sei b�sew jèsew : b-p-dimIF,b-pos-dimIF, b-ps-dimIF, b-p0-dimIF kai b-p1-dimIF. Gia k�je n 2 ! sthn kl�sh IP(df � n)up�rqoun kajolik� stoiqe�a.Apìdeixh. ProkÔptei �mesa apì thn Prìtash 3.4.8. �7.3 Kajolik� stoiqe�a gia diast�sei -sunart sei 'Estw IF mia kl�sh uposunìlwn, ID mia kl�sh p-b�sewn kai df mia apì ti diast�sei -sunart sei b�sew jèsew : b-p-dimIF, b-p0-dimIF kai b-p1-dimIF. Tìte, ìpw sto [37℄(blèpe Enìthta 3.3, sel�da 106) mporoÔme na or�soume mia kainoÔrgia di�stash-sun�rthshID-df , me ped�o orismoÔ thn kl�sh ìlwn twn q¸rwn kai ped�o tim¸n to sÔnolo ![f�1;1gw ex  : gia k�je q¸ro X, ID-df(X) e�nai to el�qisto stoiqe�o n tou ! [ f1g gia toopo�o up�rqoun èna uposÔnolo Q tou X kai mia p-b�sh B gia to Q sto X ètsi ¸ste(Q;B;X) 2 ID kai df(Q;B;X) � n.7.3.1 Sumbolismì . 'Estw df mia apì ti parak�tw diast�sei -sunart sei b�sew jèsew : b-p-dimIF, b-p0-dimIF kai b-p1-dimIF. Gia k�je n 2 f�1g [ ! me IP(ID-df � n)sumbol�zoume thn kl�sh ìlwn twn q¸rwn X me ID-df(X) � n.7.3.2 Je¸rhma. 'Estw ID koresmènh kl�sh p-b�sewn. E�n h kl�sh IF e�nai koresmènh,pl rh kai ikanopoie� ti Sunj ke Peperasmènh 'Enwsh kai KenoÔ Uposunìlou, tìtegia k�je n 2 f�1g [ ! h kl�sh IP(ID-df � n) e�nai koresmènh.Apìdeixh. 'Estw n 2 f�1g [ !. Apì to Je¸rhma 7.2.4, h kl�sh IP(df � n) e�naikoresmènh. Ja apode�xoume ìti h kl�sh IP(ID-df � n) e�nai koresmènh. 'Estw S miadiktuwmènh oikogèneia apì q¸rou pou an koun sthn kl�sh IP(ID-df � n). ApodeiknÔoumeìti up�rqei èna sun-shm�di M+ th S ètsi ¸ste gia k�je sun-shm�di M th S, pou e�naisun-epèktash touM+, na up�rqei miaM-epitrept  oikogèneia R+ apì sqèsei isodunam�a ep� th S tètoia ¸ste gia k�je epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S,pou e�nai telik¸ leptìterh th R+, kai k�je L 2 C}(R) na èqoume ìtiT(L) 2 IP(ID-df � n):

136 Kef�laio 7Gia k�je X 2 S up�rqei èna uposÔnolo QX tou X kai mia p-b�sh BX gia to QX sto Xètsi ¸ste (QX ; BX ; X) 2 ID kai df(Q;B;X) � n. Ef> ìson h kl�sh ID e�nai koresmènh,up�rqei èna arqikì sun-shm�di M+ID th S pou antistoiqe� ston periorismìQ � fQX : X 2 Sg;sth sun-diktÔwsh N � fNX : X 2 Sg th B � fBX : X 2 Sg kai sthn kl�sh ID. Ef>ìson h kl�sh IP(df � n) e�nai koresmènh, up�rqei èna arqikì sun-shm�di M+IP(df�n) th Spou antistoiqe� ston periorismì Q, sth sun-diktÔwshN th B kai sthn kl�sh IP(df � n).Sumbol�zoume me M+ èna sun-shm�di th S, to opo�o e�nai sun-epèktash twn M+ID kaiM+IP(df�n). 'Estw M mia auja�reth sun-epèktash touM+. Sumbol�zoume me R+ mia arqik okogèneia th S pou antistoiqe� sto sun-shm�di M, ston periorismì Q, sth sun-diktÔwshN th B kai sthn kl�sh ID ìpw ep�sh mia arqik  oikogèneia th S pou antistoiqe� stosun-shm�diM, ston periorismì Q, sth sun-diktÔwsh N th B kai sthn kl�sh IP(df � n).'Estw R mia auja�reth epitrept  oikogèneia R apì sqèsei isodunam�a ep� th S, h opo�ae�nai telik¸ leptìterh th R+, kai L 2 C}(R). Tìte, èqoume (T(LjQ);BL};�(�);T(L)) 2 IDkai df(T(LjQ);BL};�(�);T(L)) � n, ìpou � e�nai mia endeiktik  sun�rthsh apì to N stoM,   isodÔnama ID-df(T(L)) � n, dhlad  T(L) 2 IP(ID-df � n). Sunep¸ h kl�shIP(ID-df � n) e�nai koresmènh. �7.3.3 Pìrisma. Gia k�je n 2 ! sthn kl�sh IP(ID-df � n) up�rqoun kajolik� stoiqe�a.Apìdeixh. ProkÔptei �mesa apì thn Prìtash 3.1.7. �7.3.4 Pìrisma. 'Estw IF h kl�sh ìlwn twn zeug¸n (Q;X), ìpou Q e�nai èna anoiktìuposÔnolo enì q¸rou X, ID h kl�sh ìlwn twn p-b�sewn kai IP mia apì ti parak�twkl�sei :(1) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn me b�ro � � ,(2) h kl�sh ìlwn twn (pl rw ) kanonik¸n ountable-dimensional q¸rwn me b�ro � � ,(3) h kl�sh ìlwn twn (pl rw ) kanonik¸n strongly ountable-dimensional q¸rwn me b�ro � � ,(4) h kl�sh ìlwn twn (pl rw ) kanonik¸n lo ally �nite-dimensional q¸rwn me b�ro � �kai(5) h kl�sh ìlwn twn (pl rw ) kanonik¸n q¸rwn X me w(X) � � kai ind(X) � � 2 �+.Tìte, gia k�je n 2 ! sthn kl�sh IP(ID-df � n) \ IP up�rqoun kajolik� stoiqe�a.Apìdeixh. ProkÔptei �mesa apì thn Prìtash 3.1.6. �

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[27℄ D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Dimension-like fun tions of the typedim and universality, Topology and its Appli ations 156 (2009), no. 18, 3077{3085.[28℄ D.N. Georgiou, S.D. Iliadis, and K.L. Kozlov, The indu tive dimension of a spa eby a normal base, (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2009, no. 3,7{14; translation in Mos ow Univ. Math. Bull. 64 (2009), no. 3, 95{101.[29℄ D.N. Georgiou, S.D. Iliadis, and K.L. Kozlov, On the large trans�nite indu tivedimension of a spa e by a normal base, Mat. Vesnik 61 (2009), no. 1, 93{102.[30℄ K.P. Hart, Jun-iti Nagata and J.E. Vaughan, En y lopedia of general topology, El-sevier S ien e Publishers, B.V., Amsterdam, 2004. x+526 pp.[31℄ D.W. Henderson, D-dimension. I. A new trans�nite dimension, Pa i� J. Math.Volume 26, Number 1 (1968), 91{107.[32℄ D.W. Henderson, D-dimension. II. Separable spa es and ompa ti� ations, Pa i� J. Math. Volume 26, Number 1 (1968), 109{113.[33℄ Witold Hurewi z, Henry Wallman, Dimension Theory, Prin eton Mathemati al Se-ries, v. 4. Prin eton University Press, Prin eton, N. J., 1941. vii+165 pp.[34℄ S.D. Iliadis, A onstru tion of ontaining spa es, Topology and its Appli ations 107(2000), no 1-2, 97{116.[35℄ S.D. Iliadis, Some properties of the ontaining spa es and saturated lasses of spa es,Appl. Gen. Topol. 4 (2003), no. 2, 487{507.[36℄ S.D. Iliadis, Saturated lasses of bases, Note Mat. 22 (2003/04), no. 2, 141{156.[37℄ S.D. Iliadis, Universal spa es and mappings, North-Holland Mathemati s Studies,198. Elsevier S ien e B.V., Amsterdam, 2005. xvi+559 pp.[38℄ M. Kat�etov, P. Simon, Origins of dimension theory, Handbook of the history ofgeneral topology, Vol. 1, 113{134, Kluwer A ad. Publ., Dordre ht, 1997.[39℄ A.S. Ke hris, Classi al Des riptive Set Theory, Springer-Verlag, 1995.[40℄ R. Koga, Subspa e-dimension with respe t to total spa es, Master Thesis, OsakaKyoiku University (1998).[41℄ K. Kuratowski, Topology I, New York, 1966.139

[42℄ K. Kuratowski, Topology II, New York, 1968.[43℄ K. Kuratowski, Introdu tion to set theory and topology, Pergamon Press, 1972.[44℄ K. Kuratowski and A. Mostowski, Set Theory, With an introdu tion to des riptive settheory, Studies in Logi and the Foundations of Mathemati s, Vol. 86. North-HollandPublishing Co., Amsterdam-New York-Oxford; PWN|Polish S ienti� Publishers,Warsaw, 1976. xiv+514 pp.[45℄ M. Landau, Strong trans�nite ordinal dimension, Pro eedings of the Ameri an Ma-themati al So iety, Vol. 21, No. 3 (1969) 591{596.[46℄ H. Lebesgue, Sur la non-appli abilit�e de deux domaines appartenant respe tivement�a des espa es �a n et n+p dimensions, (Fren h) Math. Ann. 70 (1911), no. 2, 166{168.[47℄ J. van Mill, In�nite-Dimensional Topology, Prerequisites and Introdu tion, North-Holland, Amsterdam, 1989.[48℄ K. Morita, On the dimension of normal spa es I, Jap. J. Math. 20, (1950), 5{36.[49℄ K. Morita, On the dimension of normal spa es II, J. Math. So . Japan 2, (1950),16{33.[50℄ S. Mr�owka, On universal spa es, Bull. A ad. Polon. S i. Cl. III 4 (1956), 479{481.[51℄ K.R. Nagami, Dimension Theory, Volume 37 in the series Pure and Applied Mathe-mati s, A ademi Press, New York and London, 1970, 244+xi pp.[52℄ Jun-iti Nagata, Modern dimension theory, Sigma Series in Pure Mathemati s, 2.Heldermann Verlag, Berlin, 1983. ix+284 p.[53℄ Jun-iti Nagata, Topi s in dimension theory. General topology and its relations tomodern analysis and algebra, V (Prague, 1981), 497{506, Sigma Ser. Pure Math., 3,Heldermann, Berlin, 1983.[54℄ Jun-iti Nagata, On a universal n-dimensional set for metri spa es. J. Reine Angew.Math. 204 1960 132{138.[55℄ T. Nishiura, Indu tive invariants and dimension theory, Fund. Math., 59, 243{262(1966).[56℄ A.K. O'Connor, A new approa h to dimension, A ta Math. Hung. 55(1{2) (1990),83{95. 140

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Per�lhyhH kataskeu  tou Peano to 1890 mia suneqoÔ apeikìnish apì èna tm ma ep� enì tetrag¸nou èdwse aform  gia to prìblhma e�n èna tm ma kai èna tetr�gwno e�nai omoiì-morfa, kai genikìtera e�n o n-kÔbo In e�nai omoiìmorfo me ton m-kÔbo Im gia n 6= m.To prìblhma autì lÔjhke apì ton Brouwer to 1911 kai h melèth autoÔ tou probl ma-to od ghse ston orismì twn diast�sewn ind, Ind kai dim kai genikìtera sth gènesh kaian�ptuxh th Jewr�a Diast�sewn.Sth diatrib  aut  or�zontai diast�sei -sunart sei tou tÔpou ind, Ind kai dim kaiapodeiknÔontai basikè idiìthte th Jewr�a Diast�sewn (jewr mata upoq¸rou, ajro�-smato kai ginomènou) gia ti sunart sei autè . Me th bo jeia twn sunart sewn aut¸nor�zontai nèe kl�sei topologik¸n q¸rwn kai melet�tai gia ti kl�sei autè to prìblhmath kajolikìthta , dhlad  th Ôparxh   mh kajolik¸n q¸rwn gia ti kl�sei autè . 'Ena topologikì q¸ro T kale�tai kajolikì gia mia kl�sh IP topologik¸n q¸rwn, ìtan o Tan kei sthn kl�sh IP kai k�je topologikì q¸ro pou an kei sthn kl�sh IP perièqetaitopologik� sto q¸ro T . Gia thn Ôparxh kajolik¸n stoiqe�wn sti kl�sei autè qrhsi-mopoie�tai h mèjodo kataskeu  Periektik¸n Q¸rwn tou bibl�ou: S.D. Iliadis, Universalspa es and mappings, North-Holland Mathemati s Studies, 198. Elsevier S ien e B.V.,Amsterdam, 2005. xvi+559 pp.Ta apotelèsmata th diatrib  èqoun dhmosieuje� sta parak�tw periodik�:(1) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Dimension-like fun tions and uni-versality, Topology and its Appli ations 155 (2008), no. 17-18, 2196{2201.(2) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, On some new dimension-like fun -tions, Topology Pro eedings 31, no 1 (2007) pp. 125{136.(3) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, On positional dimension-like fun -tions, Topology Pro eedings 33 (2009) pp. 285{296.(4) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Positional dimension-like fun tionsof the type Ind, Est�lh gia dhmos�eush.(5) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Dimension-like fun tions of the typedim and universality, Topology and its Appli ations 156 (2009), no. 18, 3077{3085.143

Abstra tPeano' s onstru tion in 1890 of a ontinuous map of a segment onto a square gaverise to the problem of whether a segment and a square are homeomorphi and generallywhether the ubes In and Im are homeomorhi for n 6= m. This problem was solved byBrouwer in 1911 and the investigation of this problem leads to the de�nitions of ind, Ind,and dim and generally to the beginning of Dimension Theory.In this thesis we de�ne new dimension-like fun tions of the type ind, Ind and dimand we give basi properties of Dimension Theory (subspa e theorems, sum theorems,produ t theorems) for these dimension-like fun tions. Using the introdu ed dimension-like fun tions, new lasses of spa es are de�ned and the investigation of the universalityproblem for these lasses is given, that is whether there exists universal spa e in these lasses. A spa e T is said to be universal in a lass IP of spa es if T 2 IP and forevery X 2 IP there exists an embedding of X into T . For the existen e of universalelements in these lasses is used the onstru tion of Containing Spa es given in book:S.D. Iliadis, Universal spa es and mappings, North-Holland Mathemati s Studies, 198.Elsevier S ien e B.V., Amsterdam, 2005. xvi+559 pp.The results of this thesis are published in the following journals:(1) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Dimension-like fun tions and univer-sality, Topology and its Appli ations 155 (2008), no. 17-18, 2196{2201.(2) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, On some new dimension-like fun -tions, Topology Pro eedings 31, no 1 (2007) pp. 125{136.(3) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, On positional dimension-like fun -tions, Topology Pro eedings 33 (2009) pp. 285{296.(4) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Positional dimension-like fun tionsof the type Ind, submitted for publi ation.(5) D.N. Georgiou, S.D. Iliadis, and A.C. Megaritis, Dimension-like fun tions of the typedim and universality, Topology and its Appli ations 156 (2009), no. 18, 3077{3085.

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